A New Class of FBI Transforms and Applications

Abstract

We introduce a class of FBI transforms using weight functions (which includes the subclass of Sjöstrand’s FBI transforms used by Christ in (Commun Partial Differ Equ 22(3–4):359–379, 1997)) that is well suited when dealing with ultradifferentiable functions (see Definition 2.3) and ultradistributions (see Definition 2.15) defined by weight functions in the sense of Braun, Meise and Taylor (BMT). We show how to characterize local regularity of BMT ultradistributions using this wider class of FBI transform and, as an application, we characterize the BMT vectors (see Definition 1.2) and prove a relation between BMT local regularity and BMT vectors.

1 Introduction

The purpose of this paper is twofold: (i) to explore a new class of FBI transform and show that it can be used to characterize regularity in the classes of ultradifferentiable functions in the sense of Braun et al. [12]; and (ii) use these techniques to study regularity of iterated hypoelliptic constant coefficient partial differential operators. This new systematic approach has led us to a plethora of unanticipated results. We show that the FBI transforms introduced in (i) are not only fundamental to obtain the results in (ii) but also allows us to extend similar results that appeared recently where a similar program was developed with the use of the Fourier transform.

1.1 The Augmented Class of FBI Transforms

The Fourier transform can be used to characterize smoothness of distributions (Paley–Wiener Theorem) and also can be used to characterize analyticity of distributions. However, the analyticity characterization is significantly more difficult than the smoothness characterization (see [22]). An alternative tool to characterize regularity (smooth and analytic) is the FBI transform (see [4]).

In the next paragraphs we will recall several variations and generalizations of the classical FBI transform. Given $n \in \mathbb {N}$ and $0<\tau \le 1$ consider the following form in $\mathbb {R}^n \times \mathbb {R}^n$

$$\begin{aligned}F = dx_1 \wedge \dots \wedge dx_n \wedge d\left( \xi _1 +ix_1 \langle \xi \rangle ^\tau \right) \wedge \dots \wedge d\left( \xi _n +ix_n \langle \xi \rangle ^\tau \right) ,\end{aligned}$$

where $\xi =(\xi _1, \dots , \xi _n)$, $x=(x_1,\dots , x_n)$ and $\langle \xi \rangle =\sqrt{1+\sum _{j=1}^n \xi _j^2}.$ Define the function $a_{\tau }: \mathbb {R}^n \times \mathbb {R}^n \rightarrow {{\mathbb {C}}}$ with the property that

$$\begin{aligned}F = a_{\tau } (x,\xi ) dx_1 \wedge \dots \wedge dx_n \wedge d\xi _1 \wedge \dots \wedge d\xi _n.\end{aligned}$$

Using the above form in [13] M. Christ defined the following variation of the FBI transform (in the original article $\tau$ is actually $\lambda$)

$$\begin{aligned}\mathcal {F}_{\tau }u(x,\xi )= \left\langle u(x'); e^{i(x-x')\xi -\langle \xi \rangle ^\tau (x-x')}a_{\tau }(x-x',\xi )\right\rangle , \end{aligned}$$

where $(x,\xi )\in \mathbb {R}^n \times \mathbb {R}^n$, $u\in \mathcal {E}'(\mathbb {R}^n)$ (i.e., u is a compactly supported distribution) and the pairing refers to the interaction between distributions and test functions with respect to the variable $x'\in \mathbb {R}^n$. Moreover, among other interesting results he proved that for a given $s>1$ a distribution $u\in \mathcal {D}'(\mathbb {R}^n)$ is $G^s$ (Gevrey of order s, see next paragraph) in a neighborhood of $x_0$ if and only if there exist $\tau \ge \frac{1}{s}$, $v\in \mathcal {E}'(\mathbb {R}^n)$ with $v\equiv u$ in a neighborhood of $x_0$, positive constants a, C and an open neighborhood V of $x_0$ such that

$$\begin{aligned}\left| \mathcal {F}_{\tau }v(x,\xi )\right| \le Ce^{-a\Vert \xi \Vert ^{1/s}}, \quad \forall (x,\xi )\in V \times \mathbb {R}^n.\end{aligned}$$

Observe that the “limit" choice of $\tau$ to study $G^s$ regularity is $\tau = 1/s$.

In [18] we considered a more general class of FBI transform first introduced in [3] (see also [2] where this FBI transform plays a fundamental role) and we showed that it can be used to characterize Denjoy–Carleman regularity as we now explain. Consider a positive sequence $M=(M_j)$ satisfying some special properties. If $U \subset \mathbb {R}^n$ is an open set and $f\in C^{\infty }(U)$ we say that f is in ${\mathcal {E}}^M(U)$ (f is M-Denjoy–Carleman in U) if for each compact set $K \subset U$ there exist positive constants C, h such that $|\partial ^{\alpha } f(x)| \le Ch^{|\alpha |}M_{|\alpha |},$ for each $x \in K$ and $\alpha \in \mathbb {N}_0^n.$ The space of compactly supported functions in $\mathcal {E}^M(U)$ is denoted by $\mathcal {D}^M(U).$ We equip $\mathcal {E}^M(U)$ and $\mathcal {D}^M(U)$ with their usual topologies, the topological duals of these spaces are denoted by ${\mathcal {E}^M}'(U)$ and ${\mathcal {D}^M}'(U)$ respectively (see [26] for more information). The Gevrey spaces $G^s$ of order s are given by choosing $M=(j!^s)$. For a fixed sequence $M=(M_j)$ its associated function is defined by $M(t)\doteq \sup _j \log (t^j/M_j)$.

The main result in [18] can now be stated as follows: for u in ${\mathcal {E}^M}'(U)$, $0<\tau \le 1$ and $k\in \mathbb {N}$ denote

$$\begin{aligned} \mathcal {F}_{\tau }^ku(x,\xi )= \left\langle u(x'); e^{i(x-x')\xi - \Vert \xi \Vert ^\tau (x-x')^{2k}}\right\rangle , \quad (x,\xi )\in \mathbb {R}^n \times \mathbb {R}^n. \end{aligned}$$

(1.1)

Then u is $\mathcal {E}^M$ in a neighborhood of $x_0$ if and only if there exist $0<\tau \le 1$ so that $M(t)=O(t^{\tau })$ as $t\rightarrow \infty$, $v\in {\mathcal {E}^M}'(\mathbb {R}^n)$ with $v\equiv u$ in a neighborhood of $x_0$, positive constants $a,C>0$ and an open neighborhood V of $x_0$ such that

$$\begin{aligned} \left| \mathcal {F}_{\tau }^k v(x,\xi )\right| \le Ce^{-aM(\Vert \xi \Vert )}, \quad \forall (x,\xi )\in V \times \mathbb {R}^n. \end{aligned}$$

(1.2)

A natural question that arises is the following: is there a “limit" choice of $\tau$? When dealing with the Gevrey class of order s it follows that the choice is $\tau =1/s$. However this is far from being trivial when M is not a Gevrey sequence. Note that the choice ${\tau } \ge \frac{1}{s}$ is equivalent to the inclusion $G^{1/\tau }\subset G^{s}$.

Since $m(t)=t^{\tau }$ is equivalent to the associated function M of the sequence $(M_j)=(j!^{1/\tau })$, meaning there exist constants $C_1,C_2$ and $a>0$ such that $C_1M(t)\le m(t) \le C_2M(t),$ for all $t>a$ (consequently, $\mathcal {E}^M=\mathcal {E}^m$ as per Definition 2.3), a naive approach to try to answer this question is to enlarge the class of the FBI transform considered by allowing the term $\Vert \xi \Vert ^\tau$ in (1.1) to be any possible weight function as given in Definition 2.1.

In order to justify the previous sentence we need to recall the definition of ultradifferentiable functions by means of weight functions as introduced in [12] (see Definitions 2.1 and 2.3).

Recall that, given two weight functions $\omega$ and $\sigma$ the condition $\omega (t)=O(\sigma (t))$ as $t\rightarrow \infty$ implies the inclusion $\mathcal {E}^{\sigma }(\Omega )\subset \mathcal {E}^{\omega }(\Omega )$, for any open set $\Omega \subset {\mathbb {R}}^n$.

Moving on, given any ultradifferentiable class ${\mathcal {E}}^\omega$ as in Definition 2.3 the augmented class of FBI transforms announced earlier which will be suitable to study regularity problems in ${\mathcal {E}}^\omega$ would be those allowing weight functions $\sigma (\Vert \xi \Vert )$ instead of $\Vert \xi \Vert ^\tau$ in (1.2) as long as $\omega (t)=O(\sigma (t))$ as $t\rightarrow \infty$. However, the lack of regularity on general weight functions $\sigma$ make it very difficult to deal with these FBI transforms. In order to avoid this problem we will prove that for each weight function there exists an equivalent one possessing the desired regularity (see Definition 2.18 for the precise meaning of equivalence used here) and therefore define the same class of ultradifferentiable functions, see Remark 2.19.

To be more precise given a weight function $\sigma$ there exists $\mu _{\sigma } \in C^1((1,\infty ))$ such that $\sigma$ and $\mu _{\sigma }$ are equivalent (see Proposition 2.20). Let $\kappa >0$, following [13] we consider the differential form in $\mathbb {R}^n \times \mathbb {R}^n$

$$\begin{aligned}{} & {} F \doteq dx'_1 \wedge \dots \wedge dx'_n \wedge d\left( \xi _1+ix'_1 \kappa \mu _{\sigma }( \xi )\right) \wedge \dots \wedge d\left( \xi _n + i x'_n \kappa \mu _{\sigma }(\xi )\right) , \\ {}{} & {} \quad (x',\xi ) \in {\mathbb {R}}^n \times {\mathbb {R}}^n \end{aligned}$$

and the function $a^{\kappa }_{\mu _{\sigma }}:\mathbb {R}^n \times \mathbb {R}^n\rightarrow {{\mathbb {C}}}$ defined by the property

$$\begin{aligned} F=a^{\kappa }_{\mu _{\sigma }}(x',\xi ) dx'_1 \wedge \dots \wedge dx'_n \wedge d\xi _1 \wedge \dots \wedge d\xi _n. \end{aligned}$$

(1.3)

Note that, fixing $\xi \in \mathbb {R}^n$ it follows that $x' \mapsto a^{\kappa }_{\mu _{\sigma }}(x',\xi )$ is a polynomial function (consequently it is a $\mathcal {E}^{\omega }$ function). Hence, for each $u \in {\mathcal {D}^{{\sigma }}}'({\mathbb {R}}^n)$ and $\phi \in {\mathcal {D}^{{\sigma }}}({\mathbb {R}}^n)$ we can define

$$\begin{aligned}{} & {} {{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }} (\phi \,u)(x,\xi )\doteq \left\langle u_{x'}; \phi (x') \, e^{i(x-x')\cdot \xi - \kappa \mu _{\sigma }(\xi ) (x-x')^2 } a^\kappa _{\mu _{\sigma }}(x-x',\xi ) \,\right\rangle ,\nonumber \\ {}{} & {} \quad (x,\xi ) \in {\mathbb {R}}^n \times {\mathbb {R}}^n. \end{aligned}$$

(1.4)

Here the notation $u_{x'}$ is to emphasize that the ultradistribution u is acting on the function in $x'$ and the other variables are thought as parameters. When $\kappa =1$ we simply denote ${{\mathcal {F}}}_{\mu _{\sigma }}={{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }}.$

Our main result is the following FBI characterization of ultradifferentiable functions that can be viewed as a Paley-Wiener type theorem.

Theorem 1.1

Fix a weight function $\omega$ (see Definition 2.1), $\Omega \subset {\mathbb {R}}^n$ an open set and $u\in {\mathcal {D}^{\omega }}' (\mathbb {R}^n)$. In order that $u \in \mathcal {E}^{\omega }$ in a neighborhood of $x_0\in \Omega$ it is necessary and sufficient that there exist a weight function $\sigma$ with $\omega (t)=O(\sigma (t))$ as $t \rightarrow + \infty$ so that for each $\phi \in \mathcal {D}^{\sigma }(\Omega )$ there exist $C,c>0$ and a neighborhood $V\subset \Omega$ of $x_0$ such that

$$\begin{aligned} |\mathcal {F}_{\mu _{\sigma }}( \phi u) (x,\xi ) | \le C e^{-c \omega (\xi )}, \quad (x,\xi ) \in V \times \mathbb {R}^n. \end{aligned}$$

(1.5)

To prove Theorem 1.1 we actually show a slightly stronger result, see Theorem 4.2.

As it is customary in these Paley-Wiener type results one of the main ingredients to prove the sufficient part of Theorem 1.1 is the inversion formula of the FBI transform. We provide two different versions for the FBI inversion formula, see Lemmas 5.3 and 5.5. It turns out that we use the first inversion formula, Lemma 5.3, to prove the second Lemma 5.5 and this was inspired by [32, Lemma IX.4.1].

Observe that using [12, 8.9 Remark] it follows that for each sequence $(M_j)$ considered in [18] there exists a function $\omega _M$ satisfying Definition 2.1 such that the space of ultradifferentiable functions defined by $(M_j)$ coincides with the space defined by $\omega _M.$ However, given a function $\omega$ satisfying Definition 2.1 it is necessary to impose on $\omega$ an additional stronger condition (the existence of a constant $H>0$ such that $2\omega (t) \le \omega (Ht) + H,$ for each $t>0$) to obtain a sequence $(M_j)$ such that $\mathcal {E}^{M}= \mathcal {E}^{\omega }$ (see [11]). Hence, for general weight functions one can think that the local regularity characterization results presented here as natural extensions from the ones given in [18].

Previous Paley–Wiener type results for ultradifferentiable functions appeared in the literature but with the use of the Fourier transform, [12], and this is an obstacle to work with elements in their dual. The use of FBI transforms allow us to work also with ultradistributions, the natural ambient.

1.2 Application to Ultradifferentiable Vectors

As an application of Theorem 1.1 we study BMT vectors (or iterates) of constant coefficients partial differential operators $P(D)=\sum _{|\alpha |\le m}a_\alpha D^\alpha$. To be more precise recall that in [29] Nelson introduced the set of analytic vectors of a partial differential operator with analytic coefficients P(x, D) in an open set $U\subset \mathbb {R}^n$ and proved that analytic vectors are real analytic functions exactly when P(x, D) is elliptic.

Later Komatsu, in [25], as well as Kotake and Narasimhan, in [27], obtained a slightly improvement of Nelson’s result as follows: let $U \subset \mathbb {R}^n$ and P(x, D) as before then a function $f\in L^2_{loc}(U)$ is real analytic in U if and only if

• 1. $P^jf \in L^2_{loc}(U)$ (in the sense of the distributions), for each j, and

• 2. $\forall K \subset U$ compact there exist $C,h>0$ such that

  $$\begin{aligned} \Vert P^j f \Vert _{L^2(K)} \le C h^j (h j)^{jm}, \quad j \in \{0,1, \dots \}. \end{aligned}$$

  (1.6)

Here and throughout these notes $P^j f \doteq (P\circ \cdots \circ P) f$, for $j\in {\mathbb {N}},$ and $P^0 f \doteq f$.

Newberger and Zielezny initiated an investigation in the Gevrey category $G^s$ (replacing $j!^m$ by $j!^{sm}$ in (1.6)), see [30]. It is worth mentioning the work of Baouendi and Métivier [1] that deals with the case when P(D) is of principal type and hypoelliptic with analytic coefficients. They observed that if u is an s-Gevrey vector of P (in a smaller neighborhood) then there exist $s'>s$ such that u is $s'$-Gevrey. There is a vast literature concerning optimal regularity for such Gevrey vectors, $G^s$, $s\ge 1$, for instance [15, 16, 31] to cite just a few. Also, for elliptic operators a proof of the Kotake–Narasimhan theorem in some classes of ultradifferentiable functions is given in [7].

Recently, regularity problems for ultradifferentiable vectors defined by weight functions in the sense of Braun et al. [12] for constant coefficients operators (see Definition 1.2) appeared in the literature. It turns out that completeness of these spaces is equivalent to the hypoellipticity of P [23]. Additionally, there are others characterizations in terms of the decay of the Fourier transform, [5, 8, 24]. It is worth mentioning that in all of these results, the authors prove Paley–Wiener theorems only for distributions when the natural object to deal with are ultradistributions (see implication $(2) \Rightarrow (1)$ at Theorem 1.3). We define BMT-vectors as follows:

Definition 1.2

For each weight function $\omega$, open set $\Omega \subset \mathbb {R}^n$ and polynomial function $P(\xi )$ of degree m we define $\mathcal {E}^{\omega } (\Omega ;P)$ as the set of $u\in C^{\infty } (\Omega )$ such that for each compact set $K \subset \Omega$ there exists $\lambda >0$ satisfying

$$\begin{aligned} \sup _{j \in \mathbb {N}_0 } \Vert P(D)^{j}u\Vert _{L^{2}(K)} e^{-\frac{1 }{\lambda }\varphi ^*(\lambda m \, j)}< + \infty , \end{aligned}$$

where we are denoting $\varphi ^*(x)=\varphi _{\omega }^*(x) \doteq \sup \{xy-\varphi (y): \, y \ge 0\},$ for each $x>0$ (see (2.1)).

We will call the elements of $\mathcal {E}^{\omega } (\Omega ;P)$ BMT vectors.

To state our main result for ultradifferentiable vectors we need to introduce some ingredients. Let $P(D)=\sum _{|\alpha |\le m}a_\alpha D^\alpha$ be a hypoelliptic constant coefficients partial differential operator. Then, it is well known (see [21]) that there exist constants $C,K>0$ and a positive number ${\tilde{\rho }}\le 1$ such that

$$\begin{aligned} |P(\xi )|\ge C\Vert \xi \Vert ^{{\tilde{\rho }} m}, \quad \forall \Vert \xi \Vert \ge K. \end{aligned}$$

(1.7)

Define the hypoelliptic index by

$$\begin{aligned} \rho = \sup \big \{{\tilde{\rho }}\le 1\text { so that} (1.7) \text {is valid for some positive constants} C,K\big \}. \end{aligned}$$

It is known that this supremum is attained and it is a rational number. In other words there exist constants $C,K>0$ such that

$$\begin{aligned} |P(\xi )| \ge C\Vert \xi \Vert ^{\rho m}, \quad \Vert \xi \Vert \ge K. \end{aligned}$$

(1.8)

We will refer to the number $\rho$ as the Hörmander hypoellipticity index of P(D). Is is important to mention that this condition also plays an important role in [10]. For a fixed weight function $\omega$ we will define $\sigma (t)=\sigma _{\omega ,\rho }(t)=\omega (t^{\rho }).$ Note that, if $\omega$ is a weight function and $0<\rho \le 1$ then $\sigma$ is a weight function.

The next result gives a characterization of the spaces $\mathcal {E}^{\omega }(\Omega ;P)$ (see Definition 1.2) in terms of the FBI transforms introduced here. The characterization of vectors of partial differential operators using Fourier transform was first given in [9].

Theorem 1.3

Fix a weight function $\omega$, a hypoelliptic constant coefficient partial differential operator P(D) of order m on an open set $\Omega \subset {\mathbb {R}}^n$ along with his hypoellipticity index $\rho$ and define the weight function $\sigma$ as before. Let $\varphi (t)=\omega (e^t)$ and $\varphi ^*$ is Young conjugate, see (2.1). Given $u \in {\mathcal {D}^{\sigma }}' (\Omega )$ and $x_0\in \Omega$ the following conditions are equivalent:

• 1. There exists a neighborhood U of $x_0$ such that $u \in \mathcal {E}^{\omega }(U;P)$.

• 2. There exist $\phi \in \mathcal {D}^{\omega }(\mathbb {R}^n)$ (such that $\phi \equiv 1$ in a neighborhood of $x_ 0$), $C,\lambda , c>0$ and a neighborhood V of $x_0$ such that

  $$\begin{aligned}{} & {} |\mathcal {F}_{\mu _{\sigma }} \left( \phi \, [P(D)]^{N} ( u)\right) (x,\xi ) | \le C e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )} e^{-c \sigma (\Vert \xi \Vert )},\nonumber \\ {}{} & {} \quad (x,\xi ,N) \in V \times \mathbb {R}^n \times \mathbb {N}_0. \end{aligned}$$

  (1.9)

One of the main tools to prove Theorem 1.3 is a version of [20, Theorem 4.1] specialized to balls on which we obtain a better estimate (see Lemma A.1). This allows us to get rid of a strong restriction on the weight functions treated in [6, Theorem 3.3]. Specifically, in their theorem they consider weight functions $\omega$ so that $\omega \left( t^{\gamma }\right) =o(\sigma (t))$, as $t \rightarrow +\infty$, where $\gamma$ is a constant that arises from the hypoellipticity of P and $\sigma (t)=t^{1 / s}$ for some $s>1$. This is equivalent to $\omega \left( t\right) =o(t^{1/\gamma s})$, as $t \rightarrow +\infty$ which in turn, implies that $G^{s\gamma }\subset {\mathcal {E}}^\omega$.

Another important improvement is that we allow $u \in {\mathcal {D}^{\sigma }}' (\Omega )$ while in [6, Theorem 3.3] the authors considered only $u \in \mathcal {D}' (\Omega )$. Note that both, inequality (1.5) in Theorem 1.1 and condition (2) in Theorem 1.3 can be checked if u is only an ultradistribution and this will be used in a forthcoming paper to define wave front-sets for ultradistributions.

Another application is given in Corollary 6.4 where we recover the Kotake-Narasimhan theorem in this context. Also, it is important to recall that similar results for global Gevrey classes were studied in [19].

In the following we will use the notation B(x, r) to denote the ball of radius $r>0$ and centered at $x\in {\mathbb {R}}^n$. We will also use the notation ${\mathbb {N}}_0^n$ to denote the set of all multiindices $\alpha =(\alpha _1, \dots , \alpha _n)$ with $\alpha _j\in \{0,1,2,\dots \}$ for each $j\in \{1,\dots ,n\}$ and denote $|\alpha |=\alpha _1+ \cdots + \alpha _n.$

The organization of this paper is as follows: in the second chapter we recall the definitions of weight functions, ultradifferentiable functions and ultradistributions as well as we introduce the function $\mu _{\omega }$. We prove the necessity of Theorem 1.1 in chapter 3. In chapter 4 we present inversion formulas of our class of FBI transforms and we use them to prove the sufficiency of Theorem 1.1. A characterization of iterates of constant coefficients operators and a relation between ultradifferentiable functions and iterates of constant coefficients operators using our class of FBI transforms is given in chapter 5.

2 Ultradifferentiable Functions Defined by Weight Functions

This section is devoted to recalling basic definitions concerning ultradifferentiable functions and ultradistributions. First we establish the concept of weight functions used throughout this work.

Definition 2.1

A continuous function $\omega : [0, +\infty [ \rightarrow [0,+ \infty [$ with $\omega \equiv 0$ in [0, 1] and increasing in $[0,+\infty )$ is called a weight function when the following conditions are satisfied:

Moreover, given $x=(x_1, \dots , x_n) \in \mathbb {R}^n$ we will write $\omega (x)=\omega \left( \left( \sum _{j=1}^n |x_j|\right) ^{1/2} \right) .$

Example 2.2

For each $s>1$, let $\omega _s \equiv 0$ in [0, 1] and $\omega _s(t)=t^{\frac{1}{s}}-1$, for $t>1$. Note that, $\omega _s$ is a weight function.

2.1 Ultradifferentiable Functions

Before we introduce the notion of ultradifferentiable functions we need to recall the notion of Young conjugate that will be used throughout this paper. Let $\omega$ be a weight function and let $\varphi$ be defined by ($\delta$). The Young conjugate of $\varphi$, $\varphi ^*$, is defined by

$$\begin{aligned} \varphi ^*(x)=\varphi _{\omega }^*(x) \doteq \sup \{xy-\varphi (y): \, y \ge 0\}. \end{aligned}$$

(2.1)

It is important to recall that, by $\gamma$) and $\delta$), the Young conjugate of $\varphi$ is well defined. Moreover, $\varphi ^*$ is an increasing convex function, $\varphi ^*(0)=0,$ $\lim _{t \rightarrow + \infty } \frac{\varphi ^*(t)}{t} = + \infty$ and $(\varphi ^*)^*=\varphi$ (see [12, 1.3 Remark]).

Definition 2.3

Let $K \subset \mathbb {R}^n$ be a regular compact¹ set and $C^{\infty } (K)$ as in [26]. For each weight function $\omega$ and $\lambda >0$ we define

$$\begin{aligned} \mathcal {E}_{\lambda }^{\omega } (K)= \left\{ f \in C^{\infty }(K): \, \sup _{x \in K} \sup _{\alpha \in \mathbb {N}_0^n } |\partial ^{\alpha }f(x)| e^{-\frac{1 }{\lambda }\varphi ^*(\lambda |\alpha |)}< + \infty \right\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {D}_{\lambda }^{\omega } (K)= \left\{ f \in C^{\infty }(\mathbb {R}^n): \, \sup _{x \in K} \sup _{\alpha \in \mathbb {N}_0^n } |\partial ^{\alpha }f(x)| e^{-\frac{1 }{\lambda }\varphi ^*(\lambda |\alpha |)}< + \infty \,\,\,\,\text{ and } \,\,\,\, \text{ supp }\,f \subset K \right\} . \end{aligned}$$

Let $\Omega$ be an open subset of ${\mathbb {R}}^n$, we also define

$$\begin{aligned}\begin{array}{ccccccccccc} \mathcal {E}^{\omega }(\Omega )&{}=&{} {proj} &{} \text{ ind } &{} \mathcal {E}_{\lambda }^{\omega } (K)&{}\text{ and }&{}\mathcal {D}^{\omega }(\Omega )&{}=&{} {ind} &{} \text{ ind } &{} \mathcal {D}_{\lambda }^{\omega } (K).\\ &{} &{} \overleftarrow{_{ K\subset \subset \Omega }} &{}\overrightarrow{ _{\lambda>0}} &{} &{} &{} &{} &{} \overrightarrow{_{ K\subset \subset \Omega }} &{}\overrightarrow{ _{\lambda >0}} &{} \end{array} \end{aligned}$$

Remark 2.4

Let $\Omega$ be an open subset of $\mathbb {R}^n$. It follows that $\mathcal {E}^{\omega }(\Omega )$ is the set of smooth functions $f:\Omega \rightarrow \mathbb {C}$ such that for each compact set $K\subset \Omega$ there exist $\lambda , C>0$ satisfying

$$\begin{aligned} \sup _{x \in K} |\partial ^{\alpha } f(x)| \le C e^{\frac{1}{\lambda }\varphi ^*(\lambda |\alpha |)}, \quad \forall \, \alpha \in \mathbb {N}_0^n. \end{aligned}$$

And ${{\mathcal {D}}}^{\omega } (\Omega )$ is the subset of $\mathcal {E}^{\omega }(\Omega )$ consisting of all compactly supported functions.

Remark 2.5

For each open set $\Omega \subset {\mathbb {R}}^n$ condition ($\beta$) guarantees that ${{\mathcal {D}}}^{\omega } (\Omega ) \ne \{0\}$ (see [12, 2.5 Corollary]).

Example 2.6

If $s>1$ and $\omega _s$ is given as in Example 2.2, then the space $\mathcal {E}^{\omega _s}(\Omega )$ coincides with the Gevrey class of order s, $G^s(\Omega )$, for each open set $\Omega \subset {\mathbb {R}}^n$.

2.2 Technical Results on Weight Functions

This subsection is dedicated to the presentation of some technical results on weight functions and its Young conjugate. In the end of this subsection we will prove an important characterization of ultradistributions.

Throughout this section we will consider a fixed weight function $\omega$ (see Definition 2.1).

Remark 2.7

Using ($\gamma$) we see that there exists $A>1$ such that

$$\begin{aligned} 1 \le \omega (t), \quad t\ge A. \end{aligned}$$

Thus, applying ($\alpha$) it follows that

$$\begin{aligned} \omega (et)\le 2L\omega (t), \quad t\ge A. \end{aligned}$$

(2.2)

Next we will present some properties of $\varphi ^*$ (see (2.1)).

Remark 2.8

Note that, $\varphi ^*$ is an increasing function, is a convex function, $\varphi ^*(0)=0,$ $\lim _{t \rightarrow + \infty } \frac{\varphi ^*(t)}{t} = + \infty ,$ $\frac{\varphi ^*(t)}{t}$ is increasing and $(\varphi ^*)^*=\varphi$ (see [12, 1.3 Remark]).

Thus,

$$\begin{aligned}{} & {} 2 \lambda \varphi ^* \left( \frac{s+t}{2 \lambda } \right) \le \lambda \varphi ^* \left( \frac{s}{ \lambda } \right) + \lambda \varphi ^* \left( \frac{t}{ \lambda } \right) \le \lambda \varphi ^* \left( \frac{s+t}{ \lambda } \right) ,\nonumber \\ {}{} & {} \quad \text{ for } \text{ every } \lambda ,s,t>0. \end{aligned}$$

(2.3)

Remark 2.9

Using condition ($\alpha$) and [12, 1.4 Lemma] it follows that,

$$\begin{aligned} L^k \varphi ^*\left( \frac{t}{L^k} \right) + kt \le \varphi ^*(t) + \sum _{j=1}^{k} L^j,\quad t \ge 0, \quad k \in \mathbb {N}. \end{aligned}$$

Thus, for any $\lambda ,\alpha >0$ choosing $t= \alpha \lambda L^k$ we obtain

$$\begin{aligned} \frac{1}{\lambda } \varphi ^* \left( \alpha \lambda \right) +k \alpha \le \frac{1}{\lambda L^k} \varphi ^* \left( \alpha \lambda L^k \right) + \frac{1}{\lambda L^k} \sum _{j=1}^k L^j. \end{aligned}$$

(2.4)

Note that for each $h>0$ there exists $k \in \mathbb {N}$ such that $\log h<k$. Thus, considering $\alpha =j$ (for any $j \in \mathbb {N}$), it follows that

$$\begin{aligned} \frac{1 }{\lambda } \varphi ^*(j \lambda ) +j \log h \le \frac{1 }{\lambda } \varphi ^*(j \lambda ) +j k \le \frac{1}{\lambda L^k} \varphi ^*(j \lambda L^k) + \frac{1}{\lambda L^k} \sum _{q=1}^kL^q, \quad \lambda >0 \,\, \text{ and } \,\, j \in \mathbb {N}. \end{aligned}$$

Thus, denoting $C_{h}=\exp \{\frac{1}{ L^k} \sum _{q=1}^kL^q\}$ and $\lambda _{h}=\lambda L^k$, we conclude that

$$\begin{aligned} h^j e^{\frac{1 }{\lambda } \varphi ^*(j \lambda )} \le C_{h} e^{\frac{1}{ \lambda _k } \varphi ^*(j \lambda _{h})}, \end{aligned}$$

(2.5)

for each $h,\lambda >0$ and $j\in \mathbb {N}.$

Remark 2.10

If $\omega$ and $\sigma$ are weight functions such that there exists $A,C>0$ satisfying

$$\begin{aligned} \omega (x) \le C\sigma (x), \quad x\ge A; \end{aligned}$$

then there exists constants $\lambda _{\sigma }, {\tilde{L}}>0$ such that $\varphi _{\sigma }^*(t)\le \frac{1}{\lambda _{\sigma }}\varphi _{\omega }^*(\lambda _{\sigma }t)+{\tilde{L}}$, $\forall t\ge 0$.

In order to prove the above claim we will consider $L>1$ such that $\sigma (ex) \le L(\sigma (x)+1),$ for each $x>0.$ In particular,

$$\begin{aligned} \sigma (x) \ge \frac{\sigma (ex)}{L} -L \ge \frac{\frac{\sigma (e^2x)}{L} -L}{L} -L \ge \frac{\sigma (e^2x)}{L^2}-2L \ge \dots \ge \frac{\sigma (e^mx)}{L^m}-mL, \end{aligned}$$

for each $m \in \mathbb {N}$ and $x>0.$ Thus, considering m such that $e^m>A$

$$\begin{aligned} \varphi _{\sigma }^*(t)&= \sup _{y \ge 0}\{yt-\sigma (e^y) \} \le \sup _{ y \ge 0}\left\{ yt-\frac{\sigma (e^me^y)}{L^m}+mL\right\} \le \sup _{ y \ge 0}\left\{ yt-\frac{\omega (e^me^y)}{CL^m}+mL\right\} \\&\le \sup _{ y \ge 0}\left\{ yt-\frac{\omega (e^y)}{CL^m}\right\} +mL \le \frac{1}{\lambda _{\sigma }} \sup _{ y \ge 0}\left\{ yt\lambda _{\sigma }-\omega (e^y)\right\} +mL = \frac{1}{\lambda _{\sigma }}\varphi _{\omega }^*(\lambda _{\sigma }t)+mL. \end{aligned}$$

for $\lambda _{\sigma }=L^mC$ and for each $t>0.$

Remark 2.11

Since $\omega$ is increasing it follows that,

$$\begin{aligned} \omega (t)= t \frac{{\omega (t)}}{t}=t\int _{t}^{+ \infty } \frac{\omega (t)}{s^2}ds \le t\int _{t}^{+ \infty } \frac{\omega (s)}{s^2}ds, \quad t>1, \end{aligned}$$

(2.6)

hence, using ($\beta$), it follows that $\omega (t)=o(t)$ ($t \rightarrow + \infty$).

Remark 2.12

Using the definition of $\varphi$ and $\varphi ^*$, for each $b \ge 1$ and $\lambda >0$ we have,

$$\begin{aligned} \frac{1}{\lambda } \varphi ^*(\lambda b)&\ge \frac{1}{\lambda } \{ \lambda by-\varphi (y)\} = by-\frac{1}{\lambda } \omega (e^y) , \quad \forall \ y\ge 0. \end{aligned}$$

Thus, considering $x\ge 1$ and $y=\log x \ge 0,$

$$\begin{aligned} e^{\frac{1}{\lambda } \varphi ^*(\lambda b)}&\ge x^b e^{- \frac{1}{\lambda } \omega (x)}, \quad {\forall \, b\ge 1 \quad \text{ and } \quad \forall \, x \ge 0}. \end{aligned}$$

(2.7)

Moreover, since $\omega (x)=0$ (when $x \in [0,1]$), (2.7) also follows for $x \in [0,1].$ Furthermore, choosing $b=x$, it follows that, $b^b \le e^{\frac{1}{\lambda } \varphi ^*(\lambda b) +\frac{1}{\lambda } \omega (b)}.$ Thus, using Remark 2.11, there exists $b^*>0$ (independent of $\lambda$) such that

$$\begin{aligned} b^b \le e^{\frac{1}{\lambda } \varphi ^*(\lambda b)} e^{\frac{b}{\lambda } }, \quad \forall b \in [b^*;+ \infty ) \quad \lambda >0. \end{aligned}$$

Therefore, for each $\lambda >0$, there exists $C_{\lambda }$ such that

$$\begin{aligned} j!\le j^j\le C_\lambda ^{j} e^{\frac{1}{\lambda } \varphi ^*(\lambda j)}, \quad \forall j \in \mathbb {N}. \end{aligned}$$

(2.8)

Remark 2.13

Using (2.8) and (2.5) it follows that the space of all real analytic functions is contained in $\mathcal {E}^{\omega }.$

Remark 2.14

Observe that, using (2.7), for $b=n+1$, it follows that

$$\begin{aligned} \int _{\Vert \xi \Vert >r} e^{-a\omega (c \Vert \xi \Vert )} d \xi < + \infty , \end{aligned}$$

(2.9)

for each $a,c>0$ and $r\ge 1.$

2.3 Ultradistributions

Definition 2.15

The continuous dual of $\mathcal {D}^{\omega }(\Omega )$ is denoted by ${\mathcal {D}^{\omega }}^{\prime }(\Omega )$ and its elements are called ultradistributions. Also, the continuous dual of ${\mathcal {E}^{\omega }}(\Omega )$ is denoted by ${\mathcal {E}^{\omega }}^{\prime }(\Omega )$.

Remark 2.16

It follows from the definition that a linear functional $u:\mathcal {D}^{\omega }(\Omega ) \rightarrow \mathbb {C}$ is continuous if and only if u is sequentially continuous.

Lemma 2.17

Let $u: \mathcal {D}^{\omega }(\Omega ) \rightarrow \mathbb {C}$ be a linear functional. The following statements are equivalent:

• 1. u is (sequentially) continuous;

• 2. For all compact $K \subset \Omega$ and for each $\delta >0$ there exists a real positive constant $C_{\delta ,K}$ such that

  $$\begin{aligned} \left| \langle u, \phi \rangle \right| \le C_{\delta ,K} \sum _{\alpha \in \mathbb {N}_0^{n}} e^{-\delta \varphi ^*(|\alpha |/\delta )} \Vert \partial ^{\alpha } \phi \Vert _{L^{\infty }(K)}, \quad \quad \forall \phi \in \mathcal {D}^{\omega }(K) \doteq \mathcal {E}^{\omega }(\Omega ) \cap C_0^{\infty }(K).\nonumber \\ \end{aligned}$$

  (2.10)

Proof

Consider u such that (2.10) is satisfied. If $\phi _j \rightarrow 0$ in $\mathcal {D}^{\omega }(\Omega )$, then there exists a compact set $K\subset \Omega$ such that $\text{ supp } \, \phi _j \subset K$ (for all $j \in \mathbb {N}$) and there exists $\lambda >0$ such that

$$\begin{aligned} \sup _{\alpha \in \mathbb {N}_0^n} \sup _{x \in K} | \partial ^{\alpha } \phi _j(x) | \exp \left[ -\frac{1}{\lambda }\varphi ^*\left( {|\alpha |}{\lambda }\right) \right] \rightarrow 0, \quad j \rightarrow + \infty . \end{aligned}$$

Hence, for each $\epsilon >0$ there exists $j_\epsilon \in \mathbb {N}$ such that

$$\begin{aligned} \displaystyle \sup _{\alpha \in \mathbb {N}_0^n} \sup _{x \in K} | \partial ^{\alpha } \phi _j(x) | \exp \left[ -\frac{1}{\lambda }\varphi ^*\left( {|\alpha |}{\lambda }\right) \right] < \epsilon , \end{aligned}$$

when $j>j_{\epsilon }.$ Thus, for each $\delta >0$ there exists $C_{\delta ,K}>0$ such that,

$$\begin{aligned} \left| \langle u , \phi _j \rangle \right| \le C_{\delta ,K} \sum _{\alpha \in \mathbb {N}_0^n} e^{-\delta \varphi ^*(|\alpha |/\delta )} \epsilon e^{\frac{1}{\lambda } \varphi ^*(|\alpha |\lambda )}, \end{aligned}$$

where $j>j_{\epsilon }$. Considering $\delta =1/(\lambda L)$ (where L is the constant considered in ($\alpha$) and using (2.4) it follows that

$$\begin{aligned}e^{- \delta \varphi ^*(|\alpha |/\delta )} e^{\frac{1}{\lambda } \varphi ^*(|\alpha |\lambda )}= e^{- \frac{1}{\lambda L} \varphi ^*(|\alpha |\lambda L) + \frac{1}{\lambda }\varphi ^*(|\alpha |\lambda )} \le e^{A}e^{-|\alpha |}, \end{aligned}$$

where $A=A( \lambda )=\frac{1}{\lambda }$. Hence,

$$\begin{aligned}|\langle u, \phi _j \rangle | \le \epsilon C_{\delta ,K} e^A \sum _{\alpha \in \mathbb {N}_0^n} e^{-|\alpha |}, \quad j>j_{\epsilon }.\end{aligned}$$

Since $\sum _{\alpha \in \mathbb {N}_0^n} e^{-|\alpha |}<+\infty$, we can conclude that $\langle u, \phi _j \rangle \rightarrow 0.$ Therefore, u is continuous.

Conversely, suppose by contradiction that there exist a compact set $K \subset \Omega$ and $\delta >0$ such that for each $C>0$ there exists $\phi =\phi _C \in \mathcal {D}^{\omega }(K)$ where

$$\begin{aligned}|\langle u, \phi \rangle | > C \sum _{\alpha \in \mathbb {N}_0^n} e^{-\delta \varphi ^*(|\alpha |/ \delta )} \Vert \partial ^{\alpha } \phi \Vert _{L^{\infty }(K)}.\end{aligned}$$

Thus, for each $j \in \mathbb {N}$ there exists $\phi _j \in \mathcal {D}^{\omega }(K)$ such that

$$\begin{aligned} r_j \doteq |\langle u, \phi _j \rangle | > j \sum _{\alpha \in \mathbb {N}_0^n} e^{-\delta \varphi ^*(|\alpha |/ \delta )} \Vert \partial ^{\alpha } \phi _j \Vert _{L^{\infty }(K)}. \end{aligned}$$

Set $\psi _j= \phi _j/r_j.$ Note that $\text{ supp } \, \psi _j \subset K$ and $e^{-\delta \varphi ^*(|\alpha |/ \delta )} \Vert \partial ^{\alpha } \psi _j \Vert _{L^{\infty }(K)} \le \frac{1}{j}$ ($\forall \alpha \in \mathbb {N}_0^n$). Hence, ${\psi }_j \rightarrow 0$ in $\mathcal {D}^{\omega }(\Omega )$ and $|\langle u, \psi _j \rangle |=1.$ This fact means that a contradiction was obtained. Therefore, u is not continuous. $\square$

2.4 A Regular Equivalent Weight Function

Definition 2.18

Let $\omega :[0,+\infty [ \rightarrow [0,+\infty [$ and $\rho :[0,+\infty [ \rightarrow [0,+\infty [$ be functions. We say that $\omega$ and $\rho$ are equivalent ($\omega \sim \rho$) when there exist $\delta ,A,C>0$ such that $A\omega (t) \le \rho (t) \le C \omega (t),$ for $t>\delta .$

Remark 2.19

Let $\omega :[0,+\infty [ \rightarrow [0,+\infty [$ and $\rho :[0,+\infty [ \rightarrow [0,+\infty [$ be weight functions. If $\omega \sim \rho$ then ${{\mathcal {E}}}^{\omega }={{\mathcal {E}}}^{\rho }$ and ${{\mathcal {D}}}^{\omega }={{\mathcal {D}}}^{\rho }$ (see [12, 3.4 Proposition]).

The next proposition guarantees that for each weight function $\omega$ there exists a $C^1$ function $\mu$ such that $\omega \sim \mu$. Therefore, we recall that by (2.2) there exist $A,H>1$ such that $\omega (et)\le H\,\omega (t),$ when $t\ge A.$ Fix $h> \max \{ 2;\log (He)\}.$

Proposition 2.20

There exists a function $Q \in C^{1}((1, +\infty ))$ satisfying:

• 1. $\omega \sim Q$ and $\displaystyle \lim _{y \rightarrow + \infty } Q'(y)=0.$

• 2. There exist $\delta _{\omega } >0$ and $D>0$ such that, if $\mu _{\omega }(y) \doteq (h-1)Q \left( \left( \delta _{\omega } + \Vert y\Vert ^2\right) ^{1/2}\right) ,$ for $y \in \mathbb {R}^n,$ then,

  $$\begin{aligned} \mu _{\omega } (\xi )\ge & {} \omega \left( \left( \delta _{\omega } + \Vert \xi \Vert ^2\right) ^{1/2}\right) >0, \quad \forall \xi \in \mathbb {R}^n; \end{aligned}$$

  (2.11)
  $$\begin{aligned} \mu _{\omega } (\xi )\le & {} D \,\omega ((\delta _{\omega }+\Vert \xi \Vert ^2)^{1/2}), \quad \forall \xi \in \mathbb {R}^n; \end{aligned}$$

  (2.12)
  $$\begin{aligned} |\partial _{\xi _j} \mu _{\omega }(\xi )|< & {} 1, \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall \xi \in \mathbb {R}^n \,\, \text{ and } \,\, j \in \{1,\dots ,n\}. \end{aligned}$$

  (2.13)

• 3. Denoting $a^{\kappa }_{\mu _\omega }$ as in (1.3) it follows that for each compact set $K\subset \mathbb {R}^n$ there exists $C_K>0$ (independent of $\xi$) such that $\sup _{x\in K}|\partial _x^\alpha \{ a^{\kappa }_{\mu _\omega }(x,\xi )\}| \le C_{K},$ for each $\xi \in \mathbb {R}^n$ and $\alpha \in {\mathbb {N}}^n_0$.

Proof

1. Define

$$\begin{aligned} Q(y)= \int _1^{+\infty } \frac{\omega (yt)}{t^h} dt=y^{h-1} \int _{y}^{+\infty } \frac{\omega (s)}{s^{h}}ds, \quad y>0.\end{aligned}$$

Since $h>2$ it follows that $0\le \frac{\omega (s)}{s^h}\le \frac{\omega (s)}{s^2},$ when $s>1$. Recalling that $\omega \equiv 0$ in [0, 1] and using ($\beta$) we can see that the function Q is well defined. And

$$\begin{aligned} Q'(y)= (h-1)y^{h-2} \int _{y}^{+\infty } \frac{\omega (s)}{s^{h}}ds - y^{h-1} \frac{\omega (y)}{y^h}. \end{aligned}$$

Therefore, $Q'$ is a continuous function. Moreover, since $\omega$ is increasing and $h>2$ we have

$$\begin{aligned} Q(y) \ge \omega (y)\int _1^{+\infty } \frac{1}{t^h} dt = \frac{1}{h-1} \omega (y), \quad y >0. \end{aligned}$$

(2.14)

And, since $h> \log (He)$ we obtain $e^h/(He)>1.$ Consequently, $C=\sum _{j=0}^{+ \infty } \left( \frac{He}{e^{h}}\right) ^j<+\infty .$ Thus,

$$\begin{aligned} Q(y)=&\int _1^{+ \infty } \frac{\omega (yt)}{t^h}dt= \sum _{j=0}^{+ \infty } \int _{e^j}^{e^{j+1}}\frac{\omega (yt)}{t^h}dt \le \sum _{j=0}^{+ \infty } \frac{\omega (e^{j+1}y)}{e^{jh}}e^j(e-1) \end{aligned}$$

(2.15)

$$\begin{aligned} \le&\sum _{j=0}^{+ \infty } \frac{H^{j+1}\omega (y)}{e^{jh}}e^j(e-1) \le H(e-1) \omega (y)\sum _{j=0}^{+ \infty } \left( \frac{He}{e^{h}}\right) ^j =CH(e-1)\omega (y) \end{aligned}$$

(2.16)

when $y\ge A.$ Moreover, recalling that $h > 2$

$$\begin{aligned} |Q'(y)|&\le (h-1)y^{h-2} \int _{y}^{+\infty } \frac{\omega (s)}{s^{2}} s^{2-h}ds + \frac{\omega (y)}{y} \le (h-1) \int _{y}^{+\infty } \frac{\omega (s)}{s^{2}} ds + \frac{\omega (y)}{y}, \quad y>1. \end{aligned}$$

Thus using ($\beta$) and (2.6) it follows that $\lim _{y \rightarrow + \infty } |Q'(y)|=0.$

2. Since $\lim _{y \rightarrow + \infty } |Q'(y)|=0$ there is $\delta _{\omega }>A$ such that $|Q'(y)|<\frac{1}{h-1}$ when $y>\delta _{\omega }^{1/2}.$ Hence, if $\mu _{\omega }$ is as being defined above then, $|\partial _{\xi _j} \mu _{\omega }(\xi )|<1,$ for each $\xi \in \mathbb {R}^n$ and $j \in \{1, \dots , n\}.$ Moreover, using (2.14) it follows that $\mu _{\omega } (\xi )=(h-1)Q \left( \left( \delta _{\omega } + \Vert \xi \Vert ^2\right) ^{1/2}\right) \ge \omega \left( \left( \delta _{\omega } + \Vert \xi \Vert ^2\right) ^{1/2}\right) ,$ $\forall \xi \in \mathbb {R}^n.$ And, using (2.15) we obtain $\mu _{\omega } (\xi )\le D \,\omega ((\delta _{\omega }+\Vert \xi \Vert ^2)^{1/2})$ where $D=(h-1)CH(e-1).$

3. It follows from (2.13) and the definition of $a^{\kappa }_{\mu _\omega }.$ $\square$

Remark 2.21

Since $\omega$ satisfies $\delta$), it follows that the Q function defined in the proof of Proposition 2.20 also satisfies $\delta$).

Remark 2.22

The Q function defined in the proof of Proposition 2.20 was inspired by the proof of [28, 1.3. Proposition]. The function presented there is $\chi (y) = \int _1^{+\infty } \frac{\omega (yt)}{t^2} \, dt$. In their work, the authors establish a similar inequality to (2.15) by assuming a stronger hypothesis than ours: the existence of $K > 1$, $T > 0$, and $0< \epsilon < 1$ such that $\omega (Kt) \le (K - \epsilon ) \omega (t)$ for each $t \ge T$. However, since we do not have this hypothesis, we considered the power h instead of 2.

3 Technical Results on FBI Transforms

This chapter is dedicated to the presentation of some technical results. Using Faà di Bruno’s formula we will obtain estimates for derivatives of FBI terms. At the end of this section we will present a general inequality for the FBI transform acting on ultradistributions.

Lemma 3.1

Let $\omega$ be a weight function (see Definition 2.1) and $\kappa >0$. If $\mu _{\omega }$ and $\delta =\delta _{\omega }$ are defined as in Proposition 2.20 then, there exist $\delta ,D>0$ such that,

$$\begin{aligned} \left| \partial _{y}^{m}\left\{ e^{- \kappa \mu _{\omega }(\xi )(x-y)^2} \right\} \right|&\le D^{m} \theta ^{-m/2} m! e^{-\frac{\kappa }{2} \omega ((\Vert \xi \Vert ^2+\delta )^{1/2})\Vert x-y\Vert ^2} e^{\frac{\theta }{2} \omega ((\Vert \xi \Vert ^2+\delta )^{1/2})} , \end{aligned}$$

(3.1)

for all $\xi \in \mathbb {R}^n$ $x,y \in \mathbb {R}$, $m \in \mathbb {N}_0$ and $\theta >0.$

Proof

Note that, using Faà di Bruno’s formula (see [14, Corollary 2.11]) it follows that

$$\begin{aligned} \partial _{y}^{m}\left\{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right\}&= \sum _{r=1}^{m} e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \sum _{\mathfrak { p }(m,r)} m! \prod _{j=1}^m \frac{[\partial _{y}^j \{-\kappa \mu _{\omega }(\xi )(x-y)^2\}]^{k_j}}{k_j! [j!]^{k_j}}, \end{aligned}$$

where ${\mathfrak {p}}(m, r)= \left\{ (k_1, \dots , k_m) \in {\mathbb {N}}_0^m: \ \sum _{j=1}^mk_j=r, \ \sum _{j=1}^m j k_j=m \right\} .$

Since $\partial _{y}^j \{-\kappa \mu _{\omega }(\xi )(x-y)^2\}=0,$ for $j\ge 3,$ we will be able to consider the sum over a subset of $\mathfrak { p }(q,r)$, considering only derivatives of order less than three. Hence, denoting ${\mathfrak {p}}_2(m, r)= \left\{ (k_1, k_2) \in {\mathbb {N}}_0 ^2: \ k_1+k_2=r, \ k_1+2k_2=m \right\} ,$ we have

$$\begin{aligned} \left| \partial _y^{m}\left\{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right\} \right| \le&\sum _{r=1}^{m} \left| e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right| \sum _{\mathfrak { p }_2(m,r)} m! \kappa ^r \frac{ |2\mu _{\omega }(\xi )(x-y)|^{k_1}|2\mu _{\omega }(\xi )|^{k_2}}{k_1!k_2!1!^{k_1} 2!^{k_2}}. \end{aligned}$$

And, using that $r=k_1+k_2 =\frac{k_1}{2}+\frac{m}{2}$, ${k_1} \le {m}$, $m! \le 2^{m} r! k_2!$ and $k_2!k_1!^{1/2}=(k_2!^2k_1!)^{1/2} \le m!^{1/2}$ we obtain,

$$\begin{aligned} \big |\partial _{y}^{m}\big \{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \big \} \big |&\le \sum _{r=1}^{m} \kappa ^r \sum _{{\mathfrak {p}}_2(m,r)} \frac{m!2^{k_1}}{k_1!k_2! } \left[ e^{ -2\kappa \mu _{\omega }(\xi )(x-y)^2}\left| \mu _{\omega }(\xi )(x-y)^2\right| ^{k_1}\right] ^{\frac{1}{2}} |\mu _{\omega }(\xi )|^{\frac{m}{2}}\\&\le D^m\sum _{r=1}^{m} \kappa ^{\frac{m}{2}} \sum _{\mathfrak { p }_2(m,r)} \frac{m!2^{k_1}}{k_1!k_2! } \left[ e^{-2 \kappa \mu _{\omega }(\xi )(x-y)^2}e^{\frac{\kappa }{D} |\mu _{\omega }(\xi )(x-y)^2|} k_1!\right] ^{\frac{1}{2}} |\mu _{\omega }(\xi )|^{\frac{m}{2}}\\&\le [D \kappa ^{\frac{1}{2}}]^m m!^{1/2} 4^m \left[ e^{-2\kappa \mu _{\omega }(\xi )(x-y)^2}e^{ \frac{\kappa }{D}| \mu _{\omega }(\xi )|(x-y)^2} \right] ^{\frac{1}{2}} |\mu _{\omega }(\xi )|^{\frac{m}{2}}\\&\quad \times \sum _{r=1}^{m} r! \sum _{\mathfrak { p }_2(m,r)} \frac{1}{k_1!k_2!} \\&\le [D \kappa ^{\frac{1}{2}}]^m m!^{1/2} 4^m \left[ e^{-2\kappa \mu _{\omega }(\xi )(x-y)^2}e^{ \frac{\kappa }{D}| \mu _{\omega }(\xi )|(x-y)^2} \right] ^{\frac{1}{2}} |\mu _{\omega }(\xi )|^{\frac{m}{2}}\\&\quad \times \sum _{r=1}^{m} r! \sum _{\mathfrak { p }(m,r)} \prod _{j=1}^m \frac{1}{k_j!} , \end{aligned}$$

where $D>1$ is as in (2.12). Recalling that (see [14, p. 515]),

$$\begin{aligned} r! \sum _{{\mathfrak {p}}(m,r)} \prod _{j=1}^m \frac{1}{k_j!}= \left( {\begin{array}{c}m-1\\ r-1\end{array}}\right) \le \left( {\begin{array}{c}m\\ r\end{array}}\right) \end{aligned}$$

and

$$\begin{aligned} \sum _{r=0}^{m} \left( {\begin{array}{c}m\\ r\end{array}}\right) =2^{m}, \end{aligned}$$

using (2.11) and (2.12) it follows that

$$\begin{aligned} \left| \partial _{y}^{m}\left\{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right\} \right| \le [D^{\frac{3}{2}} \kappa ^{\frac{1}{2}}]^m 4^{m} m!^{1/2} e^{-\frac{\kappa }{2} \omega ((\Vert \xi \Vert ^2+\delta )^{1/2}) (x-y)^2} [\omega ((\Vert \xi \Vert ^2+\delta )^{1/2})]^{\frac{m}{2}} 2^m. \end{aligned}$$

(3.2)

Thus, for each $\theta >0$ it follows that,

$$\begin{aligned} \left| \partial _{y}^{m}\left\{ e^{-\kappa \mu _{\omega }(\xi )(z-y)^2} \right\} \right|&\le [D^{\frac{3}{2}} \kappa ^{\frac{1}{2}}16]^m \theta ^{-\frac{m}{2}} m! e^{-\frac{\kappa }{2}\omega ((\Vert \xi \Vert ^2+\delta )^{1/2})(x-y)^2} e^{(\theta /2)\omega ((\Vert \xi \Vert ^2+\delta )^{1/2})}. \end{aligned}$$

$\square$

Remark 3.2

Observe that $\partial _{y}\left\{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right\} = -\partial _{x}\left\{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right\} ,$ for $\xi \in \mathbb {R}^n$ and $y,x \in \mathbb {R}.$ Thus, we can rewrite (3.1) by

$$\begin{aligned} \left| \partial _{x}^{\ell }\partial _{y}^{m}\left\{ e^{-\kappa \mu _{\omega }(\xi )(x-y)^2} \right\} \right| \le D^{m+\ell } \theta ^{-(m+\ell )/2} (m +\ell )! e^{-\frac{\kappa }{2} \omega ((\Vert \xi \Vert ^2+\delta )^{1/2}) \,|x-y|^2} e^{(\theta /2)\omega ((\Vert \xi \Vert ^2+\delta )^{1/2})},\nonumber \\ \end{aligned}$$

(3.3)

for all $\xi \in \mathbb {R}^n$, $x,y \in \mathbb {R}$, $\ell ,m \in \mathbb {N}_0$ and $\theta >0.$

Remark 3.3

Observe that

$$\begin{aligned} \partial _{y}^{\alpha }\left\{ e^{- \kappa \mu _{\omega }(\xi )(x-y)^2} \right\} = \prod _{j=1}^n \partial _{y_j}^{\alpha _j}\left\{ e^{- \kappa \mu _{\omega }(\xi )(x_j-y_j)^2} \right\} , \end{aligned}$$

for each $x,\xi ,y=(y_1,\dots ,y_n)\in \mathbb {R}^n$ and $\alpha =(\alpha _1,\dots ,\alpha _n)\in \mathbb {N}_0.$ Thus, it follows from Lemma 3.1,

$$\begin{aligned} \left| \partial _{y}^{\alpha }\left\{ e^{- \kappa \mu _{\omega }(\xi )(x-y)^2} \right\} \right|&\le D^{|\alpha |} \theta ^{-|\alpha |/2} \alpha ! e^{-\frac{\kappa }{2} \omega ((\Vert \xi \Vert ^2+\delta )^{1/2})\Vert x-y\Vert ^2} e^{\frac{\theta n}{2} \omega ((\Vert \xi \Vert ^2+\delta )^{1/2})} , \end{aligned}$$

(3.4)

Next we will present bounds on the derivatives of the FBI phase.

Lemma 3.4

If $\omega$ is a weight function, $\mu _{\omega }$ and $\delta =\delta _{\omega }$ are defined as in Proposition 2.20, $\kappa >0$ and $Q_{\omega }^{\kappa }(z,y,\xi )=i(x-y)\cdot \xi -\kappa \mu _{\omega }(\xi ) (x-y)^2,$ then for each $\theta ,\lambda >0$ there exists $D=D_{\theta ,\lambda }$ such that

$$\begin{aligned} |\partial ^{\gamma }_x \partial ^{\alpha }_{y} \left\{ e^{Q^{\kappa }_{\omega }(x,y,\xi ) } \right\} |\le & {} D^{|\alpha | + |\gamma |} e^{\frac{1}{\lambda } \varphi ^*( \lambda |(\alpha ,\gamma )|)}\times \nonumber \\ {}{}\times & {} e^{\frac{1}{\lambda }\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})+\frac{n\theta }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})} e^{-\frac{\kappa }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})\,(x-y)^2},\nonumber \\ \end{aligned}$$

(3.5)

for $(\gamma ,\alpha ,x,y,\xi ) \in \mathbb {N}_0^n\times \mathbb {N}_0^n \times \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R}^n$.

Proof

From Leibniz rule it follows that,

$$\begin{aligned} \partial _x^{\gamma }\partial ^{\alpha }_{y} \left\{ e^{Q^{\kappa }_{\omega }(x,y,\xi ) } \right\} =&{} \!\!\!\sum \limits _{\beta \le (\gamma ,\alpha )}\!\!\! \left( {\begin{array}{c}\gamma \\ \beta '\end{array}}\right) \left( {\begin{array}{c}\alpha \\ \beta ''\end{array}}\right) (i\xi )^{\gamma -\beta '} (-i\xi )^{\alpha -\beta ''} e^{i(x-y)\xi } \\ {}{}&\times {} \prod \limits _{k=1}^{n} \partial ^{\beta '_k}_{x_k}\partial ^{\beta ''_k}_{y_k} \left\{ e^{ -\kappa \mu _{\omega }(\xi ) (x_k-y_k)^2 } \right\} , \end{aligned}$$

where $\beta =(\beta ',\beta ''),$ $\beta '=(\beta '_1, \dots ,\beta '_n)\in \mathbb {N}_0^n$ and $\beta ''=(\beta ''_1, \dots ,\beta ''_n)\in \mathbb {N}_0^n$.

Therefore, using (3.3), (2.3), (2.7) and (2.8) there exists $C,D>1$ such that,

$$\begin{aligned} |\partial _x^{\gamma }\partial ^{\alpha }_{y} \left\{ e^{Q_{\omega }(z,y,\xi ) } \right\} \!|&\le \sum _{\beta \le (\gamma ,\alpha )} \left( {\begin{array}{c}\gamma \\ \beta '\end{array}}\right) \left( {\begin{array}{c}\alpha \\ \beta ''\end{array}}\right) \Vert \xi \Vert ^{|(\gamma ,\alpha )-\beta |} D^{|\beta |} \theta ^{-\frac{|\beta |}{2}} \beta !\, e^{-\frac{\kappa }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})\,(x-y)^2} \\&\quad \times e^{\frac{n\theta }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})\,}\\&\le \sum _{\beta \le (\gamma ,\alpha )} \left( {\begin{array}{c}\gamma \\ \beta '\end{array}}\right) \left( {\begin{array}{c}\alpha \\ \beta ''\end{array}}\right) e^{\frac{1}{\lambda } \varphi ^*( \lambda |(\gamma ,\alpha )- \beta |)} e^{\frac{1}{\lambda }\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})\,} \\&\quad \times C^{|\beta |} \theta ^{-\frac{|\beta |}{2}} e^{\frac{1}{\lambda } \varphi ^*(\lambda |\beta |)} e^{\frac{|\beta |}{\lambda }} \,\\&\quad \times e^{-\frac{\kappa \omega (({\Vert \xi \Vert ^2+\delta })^{1/2})\,}{2}(x-y)^2} e^{\frac{n\theta }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})} \\&\le (1+C\theta ^{-1/2}e^{\frac{1}{\lambda }})^{|\gamma |+|\alpha |} e^{\frac{1}{\lambda } \varphi ^*( \lambda |(\gamma ,\alpha )|)} e^{\frac{1}{\lambda }\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})} \\&\quad \times e^{-\frac{\kappa \omega (({\Vert \xi \Vert ^2+\delta })^{1/2})}{2}(x-y)^2} e^{\frac{n\theta }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})} \end{aligned}$$

for each $\lambda ,\theta >0.$ $\square$

The following result consists of a general domination of the FBI transform for ultradistributions.

Lemma 3.5

Let $\Omega \subset \mathbb {R}^n$ be an open subset. If $\omega$ is a weight function, $\mu _{\omega }$ and $\delta =\delta _{\omega }$ are defined as in Proposition 2.20, $\kappa >0,$ $u \in {\mathcal {D}^{\omega }}'(\Omega )$ and $\phi \in {\mathcal {D}^{\omega }}(\Omega )$ with $K=\text{ supp }\,\phi$. Then, there exists $\lambda >0$ such that for each $\theta >0$ we can find $C=C_{\theta }>0$ such that

$$\begin{aligned} | \partial _x^{\gamma }{{\mathcal {F}}}_{\mu _{\omega }}^{\kappa } (\phi u)(x,\xi )| \le C e^{(1/\lambda ) \varphi ^*( \lambda |\gamma |)}e^{\theta \omega (({\Vert \xi \Vert ^2+\delta })^{1/2})} \sup _{y \in K}e^{ -\frac{\kappa }{2}(x-y)^2\omega (({\Vert \xi \Vert ^2+\delta })^{1/2}) }, \end{aligned}$$

for each $(x,\xi ) \in \mathbb {R}^n \times \mathbb {R}^n$ and $\gamma \in \mathbb {N}_0^n.$

Proof

Observe that denoting $e_1=(1,0, \dots ,0), \dots , e_n=(0, \dots , 0,1)$ and considering a real sequence $\{h_j\}_{j=1}^{\infty }$ such that $h_j \rightarrow 0$ it follows that, if $\psi \in \mathcal {E}^{\omega }(\mathbb {R}^n \times \mathbb {R}^n)$ is such that $\psi (x,y)=\psi _1(y)\psi _2(x,y)$ (where $\psi _1\in \mathcal {D}^{\omega }( \mathbb {R}^n)$ and $\psi _2 \in \mathcal {E}^{\omega }(\mathbb {R}^n \times \mathbb {R}^n)$), then

$$\begin{aligned} \frac{\psi (x+h_je_k,\cdot )- \psi (x,\cdot )}{h_j} \rightarrow \partial _{x_k} \psi (x,\cdot ), \end{aligned}$$

in $\mathcal {D}^{\omega },$ for each $k \in \{1, \dots n\}$ and $x \in \mathbb {R}^n$. Thus, using the continuity of u

$$\begin{aligned} \lim _{h\rightarrow 0} \frac{ \langle u_y;\psi (x+he_k,y) \rangle -\langle u_{y};\psi (x,y) \rangle }{h}&= \lim _{h \rightarrow 0}\left\langle u_y;\frac{\psi (x+he_k,y)- \psi (x,y)}{h} \right\rangle \\&= \langle u_y;\partial _{x_k}\psi (x,y) \rangle . \end{aligned}$$

Hence, it follows that

$$\begin{aligned} \partial _{x}^{\gamma }{{\mathcal {F}}}^{\kappa }_{\mu _{\omega }} (\phi u)(x,\xi )=\left\langle u(y); \phi (y) \partial _{x}^{\gamma } \left\{ e^{i(x-y)\cdot \xi - \kappa \mu _{\omega }(\xi ) (x-y)^2 } a_{\mu _{\omega }}(x-y,\xi ) \right\} \right\rangle \end{aligned}$$

where $\gamma \in \mathbb {N}_0^n.$ Moreover, from Lemma 2.17 for each $\epsilon >0$ there exists $C_{\epsilon }>0$ such that

$$\begin{aligned} | \partial _x^{\gamma }{{\mathcal {F}}}^{\kappa }_{\mu _{\omega }} u(x,\xi )| \le C_{\epsilon } \sum _{\alpha } e^{-\epsilon \varphi ^*(|\alpha |/\epsilon )} \sup _{y \in K} \left| \partial ^{\alpha }_{y} \partial _x^{\gamma } \left\{ \phi (y) e^{i(x-y)\cdot \xi - \kappa \mu _{\omega }(\xi ) (x-y)^2 } a_{\mu _{\omega }}(x-y,\xi ) \right\} \right| . \end{aligned}$$

Since $\phi \in \mathcal {D}^{\omega },$ there exists $\lambda _1>0$ such that $\Vert \partial ^{\beta ''}\phi \Vert _{\infty } \le e^{\frac{1}{2\lambda _1} \varphi ^*(2\lambda _1 |\beta ''|)},$ for each $\beta '' \in \mathbb {N}_0^n.$ Moreover, using Leibniz rule, (2.3), Lemma 3.4 and Proposition 2.20, for each $\lambda ,\theta >0$ there exist $D>0$ such that

$$\begin{aligned} | \partial _x^{\gamma } {{\mathcal {F}}}^{\kappa }_{\mu _{\omega }} u(x,\xi )|&\le C C_{\epsilon } \sum _{\alpha } e^{-\epsilon \varphi ^*\left( \frac{|\alpha |}{\epsilon }\right) } \sup _{y \in K} \sum _{\beta \le (\alpha ,\gamma )} \!\!\left( {\begin{array}{c}(\alpha ,\gamma )\\ \beta \end{array}}\right) D^{|\beta |} \\&\quad \times e^{\frac{1}{2\lambda _2} \varphi ^*(2 \lambda _2|(\alpha ,\gamma )|)+\left( \frac{1}{\lambda } +\frac{n\theta }{2} \right) \omega (({|\xi \Vert ^2+\delta })^{1/2})} e^{-\frac{\kappa }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})(x-y)^2}\\&\le C C_{\epsilon } e^{\frac{1}{\lambda _2} \varphi ^*( \lambda _2|\gamma |)} \sum _{\alpha } e^{-\epsilon \varphi ^*\left( \frac{|\alpha |}{\epsilon }\right) } e^{\frac{1}{\lambda _2} \varphi ^*( \lambda _2|\alpha |)} (1+D)^{|\alpha +\gamma |} e^{\left( \frac{1}{\lambda }+\frac{n\theta }{2}\right) \omega (({\Vert \xi \Vert ^2+\delta })^{1/2})}\\&\quad \times \sup _{y \in K} e^{-\frac{\kappa }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})(x-y)^2}, \end{aligned}$$

where $\lambda _2=\max \{\lambda ,\lambda _1\}$. Next we will consider $\epsilon = \frac{1}{\lambda _2 L^k}$, where $k \in \mathbb {N}$ is chosen such that $e^{-k} (1+D)<1$ and L is as in ($\alpha$). From, (2.4),

$$\begin{aligned} | \partial _x^{\gamma }{{\mathcal {F}}}^{\kappa }_{\mu _{\omega }} u(x,\xi )|&\le C C_{\epsilon } e^{\frac{1}{\lambda _2} \varphi ^*( \lambda _2|\gamma |)} e^{\frac{1}{\lambda L^k}\sum _{j=1}^kL^j} \sum _{\alpha } e^{-k|\alpha |} (1+D)^{|\alpha +\gamma |} e^{\left( \frac{1}{\lambda }+\frac{n\theta }{2}\right) \omega (({\Vert \xi \Vert ^2+\delta })^{1/2})}\\&\quad \times \sup _{y \in K} e^{-\frac{\kappa }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})(x-y)^2} \\&\le C' e^{\frac{1}{\lambda _2} \varphi ^*( \lambda _2|\gamma |)} (1+D)^{|\gamma |}e^{\left( \frac{1}{\lambda }+\frac{n\theta }{2}\right) \omega (({\Vert \xi \Vert ^2+\delta })^{1/2})} \sup _{y \in K} e^{-\frac{\kappa }{2}\omega (({\Vert \xi \Vert ^2+\delta })^{1/2})(x-y)^2}, \end{aligned}$$

for some $C'>0.$ From the arbitrariness of $\lambda$ and $\theta$ the proof is completed. $\square$

4 Proof Theorem 1.1: Necessity

The aim of this section is to prove the necessary condition of Theorem 1.1 i.e, when given an ultradistribution u such that u is $\mathcal {E}^{\omega }$ regular in a neighborhood of a given point $x_0$, then (1.5) is satisfied. To do so we will prove a more general result, Theorem 4.2, in which the necessity part of Theorem 1.1 will be a particular case (just choose $\alpha =0$ and $\kappa =1$ in Theorem 4.2).

The reasons to present this more general result are the following. First, we need the result for arbitrary $\alpha \ne 0$ for the FBI inversion formula for ultradifferentiable functions presented in Sect. 5.1, see Lemma 5.2 and its consequences, Lemma 5.3 and Lemma 5.5.

Second, we deal with any parameter $\kappa > 0$ since in the proof of Lemma 5.5 we consider a general $\kappa$ and use Lemma 5.3 with $\kappa /2$ (see (5.14)). For this reason it is required to allow arbitrary $\kappa > 0.$ Moreover, Lemma 5.5 is important in the proof of the sufficiency part of Theorem 1.1.

Following the main ideas of [18] we start by presenting a characterization of ultradistributions vanishing in a neighborhood of a given point. And then we will use this fact to prove the main result of this section.

Lemma 4.1

Let $\sigma$ be a weight function, $\kappa >0,$ $x_0 \in {\mathbb {R}}^n,$ $\Omega \subset \mathbb {R}^n$ be an open neighborhood of $x_0$ and $u \in {\mathcal {D}^{\sigma }}'(\Omega )$. If u vanishes in a neighborhood of $x_0,$ then for each $\phi \in \mathcal {D}^{\sigma }(\Omega )$ there exist constants $C,c,\lambda >0$ and a neighborhood $V\subset \Omega$ of $x_0$ such that

$$\begin{aligned} | \partial ^{\alpha }_x{{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }} (\phi u)(x,\xi )| \le Ce^{\frac{1}{\lambda }\varphi ^*_{\sigma }(\lambda |\alpha |)}e^{-c\sigma (\xi )}, \quad (x, \xi , \alpha ) \in V \times \mathbb {R}^n \times \mathbb {N}_0^n. \end{aligned}$$

(4.1)

Proof

Given $\phi \in {{{\mathcal {D}}}^{\sigma }}(\Omega )$ it follows that $\phi \in {{{\mathcal {D}}}^{\sigma }}(\mathbb {R}^n)$ and there exists $R>0$ such that $\text{ supp } \,\phi \subset B(x_0,R)$. Considering $u\in {{{\mathcal {D}}}^{\sigma }}'(\Omega )$ vanishing in a neighborhood of $x_0$, we can assume that there exists $0<r <R/3$ such that $u \equiv 0$ in $B(x_0,3r)$. Also, consider a function $\psi \in \mathcal {D}^{\sigma }(B(x_0,3r))$ such that $\psi \equiv 1$ in $B(x_0,2r).$ Then, ${supp} \{\phi \psi \} \subset B(x_0,3r)$ and ${supp} \{\phi (1-\psi )\} \subset B(x_0,R) {\setminus } \overline{B(x_0,r)}.$ Thus, using Lemma 3.5, there exists $\lambda >0$ such that for each $\theta >0$ there exists $C=C_{\theta }>0$ such that

$$\begin{aligned} |\partial ^{\alpha }_x {{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }}(\phi u)(x,\xi )|=&|\partial ^{\alpha }_x {{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }}(\phi (1-\psi ) u)(x,\xi )| \\ \le&Ce^{\frac{1}{\lambda }\varphi ^*_{\sigma }(\lambda |\alpha |)} e^{\theta \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} \sup _{{r< \Vert y-x_0\Vert <R}} e^{- a \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})(x-y)^2} , \end{aligned}$$

for every $(x,\xi ,\alpha ) \in \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {N}_0^n$ where $a= \frac{\kappa }{2}$. Moreover, if $\Vert x-x_0\Vert < r/2$ and $\Vert y-x_0\Vert > r$ then $\Vert x-y\Vert > r/2.$ Thus, since $\delta >0$ (here $\delta$ denotes the constant $\delta _{\sigma }$ as defined in Proposition 2.20.), recalling that $\sigma$ is an increasing function and considering $\theta =\frac{ar^2}{8}$ we obtain

$$\begin{aligned} \begin{array}{c} |\partial ^{\alpha }_x {{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }} (\phi u)(x,\xi )| \le C e^{\frac{1}{\lambda }\varphi ^*_{\sigma }(\lambda |\alpha |)}e^{-\frac{ar^2}{8}\sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} \le C e^{\frac{1}{\lambda }\varphi ^*_{\sigma }(\lambda |\alpha |)}e^{-\frac{ar^2}{8}\sigma (\xi )},\\ \text{ for } \text{ all } \ \Vert x-x_0\Vert <{\frac{r}{2}}, \xi \in \mathbb {R}^n \ \text{ and } \ \alpha \in \mathbb {N}_0^n. \end{array} \end{aligned}$$

Thus setting $V=B(x_0,{r/2})$ finishes the proof. $\square$

Next we shall present the main result of this section whose consequence is the necessity part of Theorem 1.1. Note that if $\sigma$ and $\omega$ are weight functions with $\omega (t)=O(\sigma (t))$ as $t \rightarrow + \infty$ then $\mathcal {D}^{\sigma }(\Omega )\subset \mathcal {D}^{\omega }(\Omega )$ (see Remark 2.10) and ${\mathcal {D}^{\omega }}' (\Omega )\subset {\mathcal {D}^{\sigma }}' (\Omega )$.

Theorem 4.2

Let $\omega$ and $\sigma$ be weight functions such that $\omega (t)=O(\sigma (t))$ as $t \rightarrow + \infty$, $\kappa >0$, $\Omega \subset {\mathbb {R}}^n$ be an open set and $u \in {\mathcal {D}^{\sigma }}' (\Omega )$. If u is $\mathcal {E}^{\omega }$ in a neighborhood of $x_0\in \Omega$ then for each $\phi \in \mathcal {D}^{\sigma }(\Omega )$ there exist $C,c, \lambda >0$ and a neighborhood $V\subset \Omega$ of $x_0$ such that

$$\begin{aligned} |\partial _x^{\alpha }\mathcal {F}^{\kappa }_{\mu _{\sigma }}( \phi u) (x,\xi ) | \le C e^{\frac{1}{\lambda }\varphi ^*(\lambda |\alpha |)}e^{-c \omega (\xi )}, \quad (x,\xi , \alpha ) \in V \times \mathbb {R}^n \times \mathbb {N}_0^n, \end{aligned}$$

(4.2)

where $\varphi ^*=\varphi _{\omega }^*$ is the Young conjugate of $\varphi (t)=\omega (e^t).$

Proof

Fix $R>0$ such that $u\in \mathcal {E}^{\omega }(B(x_0,R))$ and $B(x_0,R) \subset \Omega$. Consider $\psi \in \mathcal {D}^{\sigma }(B(x_0,R))$ such that $\psi \equiv 1$ in $B(x_0, \tfrac{R}{2})$ and write

$$\begin{aligned} \mathcal {F}^{\kappa }_{\mu _{\sigma }}( \phi u) (x,\xi )=\mathcal {F}^{\kappa }_{\mu _{\sigma }}( \psi \phi u) (x,\xi )+\mathcal {F}^{\kappa }_{\mu _{\sigma }}( (1-\psi )\phi u) (x,\xi ), \quad (x,\xi ) \in \mathbb {R}^n \times \mathbb {R}^n.\nonumber \\ \end{aligned}$$

(4.3)

Now we use Lemma 4.1 to obtain constants $C,a,\lambda >0$ and a bounded neighborhood V of $x_0$ such that

$$\begin{aligned} |\partial _x^{\alpha } {{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }} ((1-\psi ) \phi u)(x,\xi )| \le C e^{\frac{1}{\lambda }\varphi _\sigma ^*(\lambda |\alpha |)} e^{-a\sigma (\xi )}, \quad (x, \xi ) \in V \times \mathbb {R}^n. \end{aligned}$$

Furthermore, since $\omega (t)=O(\sigma (t))$ as $t\rightarrow + \infty$ there exist $A,c>0$ so that $\omega (\xi )\le c\sigma (\xi )$ for $\Vert \xi \Vert >A$ and by the continuity of $\omega$, for each $c>0$ there exists $C_c>0$ such that $1= e^{c\omega (\xi )} e^{-c\omega (\xi )} \le C_c e^{-c\omega (\xi )}$ for $\Vert \xi \Vert \le A$. Thus, we use Remark 2.10, define $a_1= \frac{a}{c}$, increase $\lambda , C>0,$ if necessary, and obtain

$$\begin{aligned} |\partial _x^{\alpha } {{\mathcal {F}}}^{\kappa }_{\mu _{\sigma }} ((1-\psi ) \phi u)(x,\xi )| \le C e^{\frac{1}{\lambda }\varphi ^*(\lambda |\alpha |)} e^{-a_1\omega (\xi )}, \quad (x, \xi ) \in V \times \mathbb {R}^n. \end{aligned}$$

(4.4)

Next we will fix $\xi =(\xi _1, \dots , \xi _n) \in \mathbb {R}^n{\setminus } \{0\}$ and consider $j\in \{1, \dots , n\}$ such that $\Vert \xi \Vert \le n |\xi _j|$. Moreover, given an arbitrary $N \in \mathbb {N}_0$, integrating by parts with respect to $y_j$, we can write

$$\begin{aligned} \mathcal {F}^{\kappa }_{\mu _{\sigma }}( \psi \phi u) (x,\xi ) = \frac{(-1)^N}{(-i\xi _j)^N } \int e^{i(x-y)\xi } \partial _{y_j}^N \left\{ u(y)\psi (y) \phi (y) e^{-\kappa \mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right\} dy, \end{aligned}$$

for $x,\xi \in \mathbb {R}^n,$ where it was used that $\psi \phi u \in \mathcal {D}^{\omega }(B(x_0,R))\subset \mathcal {D}^{\omega }(\mathbb {R}^n)$. Given $\alpha \in \mathbb {N}_0^n$, we have

$$\begin{aligned} \partial _x^{\alpha }\mathcal {F}^\kappa _{\mu _{\sigma }}( \psi \phi u) (x,\xi ) =&\frac{(-1)^N}{(-i\xi _j)^N } \int \sum _{\gamma \le \alpha } \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) (i\xi )^{\alpha -\gamma }e^{i(x-y)\xi } \times \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \partial _x^{\gamma }\partial _{y_j}^N \left\{ u(y)\psi (y) \phi (y) e^{-\kappa \mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right\} dy \end{aligned}$$

(4.5)

Now, Remark 2.8, (2.5), (2.8) together with Leibniz’s rule and Proposition 2.20 (where it is proved that the derivatives of $a_{\mu _{\sigma }}$ are uniformly bounded in compact sets) show that there exist $C_2,\lambda _2>0$ (independent of x and $\xi$) such that

$$\begin{aligned} \begin{array}{c} {|\partial _x^{\beta _1} \partial ^{\beta _2}_y\left\{ u(y) \psi (y) \phi (y) a_{\mu _{\sigma }}(x-y,\xi )\right\} | \le C_2 e^{\frac{1 }{\lambda _2}\varphi ^*(\lambda _2 |\beta _2|)},}\\ \forall \beta _1,\beta _2 \in \mathbb {N}_0^n \ \ \text{ and } \ \ \forall \, y,\xi \in \mathbb {R}^n \ \text{ and }\ x \in V. \end{array} \end{aligned}$$

(4.6)

Furthermore, it follows from inequality (3.2) that there exist $D>0$ such that

$$\begin{aligned} \begin{array}{l} \left| \partial _x^{\beta _1}\partial _{y}^{\beta _2}\left\{ e^{- \kappa \mu _{\sigma }(\xi )(x-y)^2} \right\} \right| =\left| \prod _{j=1}^n \partial _{x_j}^{\beta ^j_1}\partial _{y_j}^{\beta ^j_2}\left\{ e^{-\kappa \mu _{\sigma }(\xi )(x_j-y_j)^2} \right\} \right| \\ \hspace{0.5cm} \le D^{|\beta _1|+|\beta _2|} e^{- \frac{\kappa }{2}{\sigma }( ({\Vert \xi \Vert ^2 + \delta _{\sigma }})^{1/2})(x-y)^2}[{\sigma }( ({\Vert \xi \Vert ^2 + \delta _{\sigma }})^{\frac{1}{2}})]^{\frac{|\beta _1+\beta _2|}{2}} |\beta _1+\beta _2|!^{\frac{1}{2}},\\ \hspace{1cm}\text{ for } \text{ all } \ \beta _1,\beta _2 \in \mathbb {N}_0^n \ \text{ and } \ x,y,\xi \in \mathbb {R}^n. \end{array} \end{aligned}$$

(4.7)

Denote $\xi _{\sigma }\doteq {\sigma }( ({\Vert \xi \Vert ^2 + \delta _{\sigma }})^{1/2})$, increase D (if necessary) and use (2.7) to obtain

$$\begin{aligned}&\left| \partial _x^{\beta _1}\partial _{y}^{\beta _2}\left\{ e^{-\kappa \mu _{\sigma }(\xi )(x-y)^2} \right\} \right| \le D^{|\beta _1+\beta _2|}|\beta _1|^{\frac{|\beta _1|}{2}}|\beta _2|^{\frac{|\beta _2|}{2}} (\xi _{\sigma } ^{|\beta _1+\beta _2|})^{\frac{1}{2}}\nonumber \\ {}&\quad \le e^{\frac{\varphi ^*(\lambda _0|\beta _1|)}{2\lambda _0}} e^{\frac{1}{2\lambda _0}\omega ( D^2|\beta _1|)} e^{\frac{\varphi ^*(\lambda _1|\beta _2|)}{2\lambda _1}} e^{\frac{1}{2\lambda _1}\omega (D^2 |\beta _2|)}e^{\frac{\varphi ^*(\lambda _1|\beta _2|)}{2\lambda _1}} e^{ \frac{1}{2\lambda _1} \omega (\xi _{\sigma })} e^{\frac{\varphi ^*(\lambda _0|\beta _1|)}{2\lambda _0}} e^{ \frac{1}{2\lambda _0} \omega (\xi _{\sigma })} ,\nonumber \\ {}&\qquad \text{ for } \text{ all } \ \lambda _0,\lambda _1>0, \ \beta _1,\beta _2 \in \mathbb {N}_0^n \ \text{ and } \ x,y \in \mathbb {R}^n.. \end{aligned}$$

(4.8)

Using Remark  2.11 we can find $D_1>D$ such that $\omega (t) \le t,$ for each $t>D_1^2$ and since $\omega$ is increasing we can write

$$\begin{aligned} \begin{array}{c} \left| \partial _x^{\beta _1} \partial _{y}^{\beta _2}\left\{ e^{-\kappa \mu _{\sigma }(\xi )(x-y)^2} \right\} \right| \le e^{\frac{\varphi ^*(\lambda _0|\beta _1|)}{\lambda _0}} e^{\frac{\varphi ^*(\lambda _1|\beta _2|)}{\lambda _1}} e^{\frac{D_1^2|\beta _1|}{2\lambda _0} } e^{\frac{D_1^2 |\beta _2|}{2\lambda _1} } e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })} ,\\ \text{ for } \text{ all } \ \lambda _0,\lambda _1>0, \ \beta _1,\beta _2 \in \mathbb {N}_0^n \ \text {and} \ x,y\in \mathbb {R}^n. \end{array} \end{aligned}$$

(4.9)

Thus, using Leibniz’s rule, Remark 2.8, (4.6) and (4.9) there exists $m=m(n)>0$ such that

$$\begin{aligned}&\Big |\partial _x^{\beta _1}\partial _{y}^{\beta _2} \Big \{u(y)\psi (y) \phi (y) e^{-\kappa \mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \Big \}\Big | \nonumber \\ {}&\quad \le \sum _{\gamma \le \beta } \left( {\begin{array}{c}\beta \\ \gamma \end{array}}\right) C_2e^{\frac{1 }{\lambda _2}\varphi ^*(\lambda _2 |\beta _2-\gamma _2|)} e^{\frac{\varphi ^*(\lambda _0|\gamma _1|)}{\lambda _0}} e^{\frac{\varphi ^*(\lambda _1|\gamma _2|)}{\lambda _1}} e^{\frac{D_1^2|\gamma _1|}{2\lambda _0} } e^{\frac{D_1^2 |\gamma _2|}{2\lambda _1} } e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })}\nonumber \\ {}&\quad \le {C_2}\,e^{\frac{1}{\lambda _1}\varphi ^*(\lambda _1 |\beta _2|)}e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\beta _1|)} \left( m+me^{\frac{D_1^2}{2\lambda _0}} \right) ^{|\beta _1|} \left( m+me^{\frac{D_1^2}{2\lambda _1}} \right) ^{|\beta _2|} e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })}\nonumber \\ {}&\quad \le C_2\,e^{\frac{1}{\lambda _1}\varphi ^*(\lambda _1 |\beta _2|)}e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\beta _1|)} C_{\lambda _0}^{|\beta _1|} C_{\lambda _1}^{|\beta _2|} e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })}, \nonumber \\ {}&\qquad \text{ for } \text{ all } \ \lambda _0>0, \ \beta _1,\beta _2 \in \mathbb {N}_0^n, x\in V \ \text{ and } \ y\in \mathbb {R}^n, \end{aligned}$$

(4.10)

where we considered $\lambda _1=\lambda _2$, used the notation $\beta =(\beta _1,\beta _2), \gamma =(\gamma _1,\gamma _2) \in \mathbb {N}_0^n \times \mathbb {N}_0^n$ and denoted $C_{\lambda }=m+me^{\frac{D_1^2}{2\lambda }}$ (for $\lambda >0$).

Hence, recalling that $\Vert \xi \Vert \le n|\xi _j|$, denoting ${\tilde{C}}=C_2 \,|\text{ supp }\, \psi \, \phi |$ (where $\,|\text{ supp }\, \psi \, \phi |=\int _{\text{ supp }\, (\psi \, \phi )}1\,dy$) and using (4.10) one can estimate the expression in (4.5) by

$$\begin{aligned}&\left| \partial _x^{\alpha }\mathcal {F}^{\kappa }_{\mu _{\sigma }}(\psi \phi u) (x,\xi ) \right| \le \frac{{\tilde{C}}n^N}{\Vert \xi \Vert ^N } \sum _{\gamma \le \alpha } \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) C_2 \Vert \xi \Vert ^{|\alpha -\gamma |}\,e^{\frac{1}{\lambda _1}\varphi ^*(\lambda _1 N)}e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\gamma |)} C_{\lambda _0}^{|\gamma |} C_{\lambda _1}^{N} e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })} \nonumber \\ {}&\quad \text{ for } \text{ all } \ N \in \mathbb {N}_0, \lambda _0>0, x\in V \ \text{ and } \ \xi \in \mathbb {R}^n \setminus \{0\} . \end{aligned}$$

(4.11)

Denoting $a=\frac{1}{nC_{\lambda _1}}$ and using Remark 2.8, (2.7) and (2.3) there exists $k_{\lambda _0}=k(\lambda _,a)>0$ such that

$$\begin{aligned} \left| \partial _x^{\alpha }\mathcal {F}^{\kappa }_{\mu _{\sigma }}(\psi \phi u) (x,\xi ) \right| \le&{\tilde{C}} \frac{1}{(a\Vert \xi \Vert )^N } k_{\lambda _0}^{|\alpha |} \,e^{\frac{1}{\lambda _1}\varphi ^*(\lambda _1 N)} e^{\frac{1}{\lambda _0}\omega (a\Vert \xi \Vert )}e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\alpha |)} e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })} \\ {}&\text{ for } \text{ all } \ \lambda _0>0, \ x\in V \ \text{ and } \ \xi \in \mathbb {R}^n \setminus \{0\} . \end{aligned}$$

Taking the infimum in $N \in \mathbb {N}_0$ in the last inequality we obtain

$$\begin{aligned} \left| \partial _x^{\alpha }\mathcal {F}^{\kappa }_{\mu _{\sigma }}(\psi \phi u) (x,\xi ) \right| \le&{\tilde{C}}\inf _{N \in \mathbb {N}_0} \frac{e^{\frac{1}{\lambda _1}\varphi ^*(\lambda _1 N)}}{(a\Vert \xi \Vert )^N } k_{\lambda _0}^{|\alpha |}\,e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\alpha |)} e^{\frac{1}{\lambda _0}\omega (a\Vert \xi \Vert )} e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })} \nonumber \\ {}&\text{ for } \text{ all } \ \lambda _0>0, \ x\in V \ \text{ and } \ \xi \in \mathbb {R}^n \setminus \{0\} . \end{aligned}$$

(4.12)

Recalling that $\inf _{N \in \mathbb {N}_0} t^{-N} e^{\frac{1}{\lambda _1} \varphi ^*({\lambda _1}N)} \le e^{\log t}e^{-\frac{1}{\lambda _1}\omega (t)}$ for each $t\ge 1$, see [12, page 218],then by increasing ${\tilde{C}}$ if necessary, it follows that one can further estimate the expression in (4.12) by

$$\begin{aligned} \left| \partial _x^{\alpha }\mathcal {F}^{\kappa }_{\mu _{\sigma }}(\psi \phi u) (x,\xi ) \right| \le&{\tilde{C}} e^{\log (a\Vert \xi \Vert )-\frac{1}{\lambda _1} \omega (a\Vert \xi \Vert )} k_{\lambda _0}^{|\alpha |} e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\alpha |)} e^{\frac{1}{\lambda _0}\omega (a\Vert \xi \Vert )} e^{\left( \frac{1}{2\lambda _0} + \frac{1}{2\lambda _1}\right) \omega (\xi _{\sigma })}\\ {}&\text{ for } \text{ all } \ \lambda _0>0, \ x\in V \ \text{ and } \ \Vert \xi \Vert \ge 1. \end{aligned}$$

Now, we take advantage of $\gamma$) and Remark 2.11 to obtain $A_1>0$ such that

$\xi _{\sigma }=\sigma (({ \Vert \xi \Vert ^2+\delta _\sigma })^{1/2}) \le {a}\Vert \xi \Vert$ and $\log (a\Vert \xi \Vert ) \le \frac{1}{{4}\lambda _1}\omega (a\xi ),$ when $\Vert \xi \Vert >A_1$.

Thus, choosing $\lambda _0=12\lambda _1$

$$\begin{aligned} \left| \partial _x^{\alpha }\mathcal {F}^\kappa _{\mu _{\sigma }}( \psi \phi u) (x,\xi ) \right|&\le {\tilde{C}} e^{-\frac{1}{8\lambda _1} \omega (a\Vert \xi \Vert )} k_{\lambda _0}^{|\alpha |}\,e^{\frac{1}{\lambda _0}\varphi ^*(\lambda _0 |\alpha |)}, \end{aligned}$$

(4.13)

for each $x \in V$ and $\Vert \xi \Vert > A_1.$ Now, we choose $k \in \mathbb {N}$ satisfying $e^{-k}\le a$. Hence, using (2.2) we see that there exists $A_2\ge A_1$ such that

$$\begin{aligned} -\omega ({a}\xi ) \le - \omega (e^{-k} \xi ) \le -(2L)^{-k} \omega (\xi ), \quad \Vert \xi \Vert >A_2. \end{aligned}$$

Thus, using (2.5) there exist $a_1>0$ and $\lambda _3>0$ such that one can further estimate (4.13) by

$$\begin{aligned} \left| \partial _x^{\alpha }\mathcal {F}^\kappa _{\mu _{\sigma }}( \psi \phi u) (x,\xi ) \right|&\le {\tilde{C}} e^{-a_1\omega (\xi )} e^{\frac{1}{\lambda _3}\varphi ^*\left( \lambda _3|\alpha |\right) }, \quad \forall x \in V,\,\, \Vert \xi \Vert >A_2 . \end{aligned}$$

(4.14)

This concludes the proof when $\Vert \xi \Vert >A_2$. For $\xi$ small, we go back to (4.5) and use (4.10) to get (possibly by increasing m)

$$\begin{aligned} |\partial _x^{\alpha }\mathcal {F}^\kappa _{\mu _{\sigma }}( \psi \phi u) (x,\xi )| =&\left| \sum _{\gamma \le \alpha } \! \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) \! \int \!\! e^{i(x-y)\xi } \partial _x^{\gamma }\partial _{y}^{\alpha -\gamma } \left\{ u(y)\psi (y) \phi (y) e^{-\kappa \mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right\} \! dy\right| \\ \le&{\tilde{C}} e^{\frac{1 }{\lambda _1}\varphi ^*(\lambda _1 (|\alpha |))} \left( m+me^{\frac{D_1^2}{2\lambda _1}} \right) ^{|\alpha |} e^{\frac{1}{2\lambda _1} \omega ( \xi _{\sigma })}. \end{aligned}$$

Since $\omega$ and $\sigma$ are continuous function (consequently $\sup _{\Vert \xi \Vert \le A_2} e^{\frac{1}{2\lambda _1} \omega (\xi _\sigma )}<+\infty$), using (2.5), there exists $\lambda _2, \,{\tilde{C}}_1, \, {\tilde{C}}_2>0$ such that

$$\begin{aligned} |\partial _x^{\alpha }\mathcal {F}^\kappa _{\mu _{\sigma }}( \psi \phi u) (x,\xi )| \le&{\tilde{C}}_1 e^{\frac{1 }{\lambda _2}\varphi ^*(\lambda _2 (|\alpha |))} \le {\tilde{C}}_2 e^{\frac{1 }{\lambda _2}\varphi ^*(\lambda _2 (|\alpha |))} e^{-a_1\omega (\xi )} \end{aligned}$$

(4.15)

for $\Vert \xi \Vert \le A_2,$ ${\tilde{C}}_2 \ge {\tilde{C}}_1 \sup _{\Vert \xi \Vert \le A_2} e^{a_1\omega (\xi )}.$ Therefore, it follows from (4.3), (4.4), (4.14) and (4.15) that the proof is completed. $\square$

An immediate consequence is the necessity of Theorem 1.1.

Corollary 4.3

Fix a weight function $\omega$, an open set $\Omega \subset {\mathbb {R}}^n$ and $u\in {\mathcal {D}^{\omega }}' (\Omega )$. If $u \in \mathcal {E}^{\omega }$ in a neighborhood of $x_0\in \Omega$ then there exist a weight function $\sigma$ such that $\omega (t)=O(\sigma (t))$ as $t \rightarrow + \infty$, so that for each $\phi \in \mathcal {D}^{\sigma }(\Omega )$ there exist $C,c>0$ and a neighborhood $x_0\in V\subset \Omega$ such that

$$\begin{aligned} |\mathcal {F}_{\mu _{\sigma }}( \phi u) (x,\xi ) | \le C e^{-c \omega (\xi )}, \quad (x,\xi ) \in V \times \mathbb {R}^n. \end{aligned}$$

(4.16)

Proof

Under the corollary hypothesis one can select any weight function $\omega (t)=O(\sigma (t))$ as $t \rightarrow + \infty$ then the proof of Theorem 4.2 will work for $u\in {\mathcal {D}^{\omega }}' (\Omega )\subset {\mathcal {D}^{\sigma }}' (\Omega )$. $\square$

5 Proof Theorem 1.1: Sufficiency

In order to prove the sufficiency part of Theorem 1.1 we shall prove an inversion formula for the FBI transform $\mathcal {F}^{\kappa }_{\mu _{\omega }}$ defined in (1.4).

5.1 Inversion Formula for Ultradifferentiable Functions

Throughout this section we will consider a weight function $\omega$ and we will denote by $\mu _{\omega }$ the function defined in Proposition 2.20. In order to prove the sufficient condition of Theorem 1.1 in this section we will present two inversion formulas of ${{\mathcal {F}}}^{\kappa }_{\mu _{\omega }}u$ when u is in $\mathcal E^{\omega }$, $\kappa >0$ and $\omega$ is any weight function.

Lemma 5.1

If $u \in C_c^{n+1}({\mathbb {R}}^n)$ then

$$\begin{aligned} u(x)=(2\pi )^{-n} \int _{{\mathbb {R}}^n} {{\mathcal {F}}}^{\kappa }_{ \mu _{\omega }} u (x,\xi ) d\xi , \forall x \in {\mathbb {R}}^n. \end{aligned}$$

Proof

Let $R>1$ be such that $\text{ supp }\,u \subset B(0,R).$ Since $u \in C_c^{n+1}({\mathbb {R}}^n)$ it follows that $\sup _{\xi \in {\mathbb {R}}^n} \{(1+\Vert \xi \Vert )^{n+1} |{\hat{u}}(\xi )| \}< + \infty$. Thus, using the Fourier inversion formula we can write

$$\begin{aligned} u(x) =&\,(2 \pi )^{-n} \lim _{\epsilon \rightarrow 0} \int \!\! \int u(x')e^{i(x-x') \cdot \xi -\epsilon \xi ^2 } d \xi dx' . \end{aligned}$$

(5.1)

Define $\Gamma = \Gamma (x,x', \xi , t) \doteq \xi +i\, t \, \kappa \, \mu _{\omega }(\xi )\, (x-x'),$ for $x,x',\xi \in \mathbb {R}^n$ and $t \in \mathbb {R}$. Since $\mu _{\omega } \ge 0$,

$$\begin{aligned} \begin{array}{c} \mathcal {R}\{i(x-x') \cdot \Gamma -\epsilon \cdot \Gamma ^2 \} \le - \epsilon \left[ \xi ^2 -t^2 \, \kappa ^2 \, [\mu _{\omega }(\xi ) ]^2\Vert x-x'\Vert ^2\right] ,\\ \text{ for } \text{ all } \ (x,x',\xi ,t) \in {\mathbb {R}}^{n} \times {\mathbb {R}}^n \times {\mathbb {R}}^n \times [0,1] . \end{array} \end{aligned}$$

This, together with the fact that the function $\omega$ is increasing and (2.12) we see that there exist constants $D,\delta >0$ such that

$$\begin{aligned}&\mathcal {R}\{i(x-x') \cdot \Gamma (x,x', \xi , t) -\epsilon \cdot [\Gamma (x,x', \xi , t)]^2 \}\\&\qquad {\le } - \epsilon \left( \xi ^2 -\kappa ^2 {[D\omega (\Vert \xi \Vert +\delta ^\frac{1}{2})]^2}|(R+r)^2 \right) ,\\&\qquad \qquad \text{ for } \ 0\le t \le 1, \Vert x'\Vert \le R, \Vert x\Vert <r \ \text{ and } \ \xi \in \mathbb {R}^n, \end{aligned}$$

for an arbitrary $r>0.$

Moreover, using (2.6) there exists a positive constant ${\tilde{A}}$ such that one can further estimate the last expression as follows

$$\begin{aligned}&\mathcal {R}\{i(x-x') \cdot \Gamma (x,x', \xi , t) -\epsilon \cdot [\Gamma (x,x', \xi , t)]^2 \} \le - \epsilon \left( 1 - \frac{1}{2} \right) \Vert \xi \Vert ^2 =- {\frac{\epsilon }{2}} \Vert \xi \Vert ^2 ,\nonumber \\ {}&\qquad \text{ for } \ 0\le t \le 1, \ \Vert x'\Vert \le R, \ \Vert x\Vert <r \ \text{ and } \ \Vert \xi \Vert >{\tilde{A}}. \end{aligned}$$

(5.2)

Since $\xi \mapsto e^{- \epsilon {\tilde{a}} \Vert \xi \Vert ^2}$ is a $L^1$ function for each $\epsilon >0$, using (5.2) it follows that

$$\begin{aligned} u(x) =&\,(2 \pi )^{-n} \lim _{\epsilon \rightarrow 0} \lim _{S \rightarrow + \infty } \int _{{\mathbb {R}}^n} \! \int _{\Vert \xi \Vert \le S} u(x')e^{i(x-x') \cdot \xi -\epsilon \xi ^2 } d \xi dx' . \end{aligned}$$

Now, using that $\zeta \mapsto u(x')e^{i(x-x') \cdot \zeta -\epsilon \zeta ^2 }$ is holomorphic, we may apply Stokes’ theorem together with the definition of $a_{\mu _{\omega }}$ (see (1.3)) to obtain,

$$\begin{aligned} u(x)=(2 \pi )^{-n} \lim _{\epsilon \rightarrow 0^+} \lim _{S \rightarrow + \infty } \int _{{\mathbb {R}}^n} \! \int _{\Vert \xi \Vert \le S} \!\!\!\! u(x')e^{i(x-x') \cdot \Gamma (x,x', \xi , 1)-\epsilon [\Gamma (x,x', \xi , 1)]^2 } a^{\kappa }_{ \mu _{\omega }}(x-x',\xi )d \xi dx'.\nonumber \\ \end{aligned}$$

(5.3)

Thus, as a consequence of the third item in Proposition 2.20, inequality (5.2) and Fubini’s theorem, we can further express (5.3) as

$$\begin{aligned} u(x)&=(2 \pi )^{-n} \lim _{\epsilon \rightarrow 0^+} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n} u(x')e^{i(x-x') \cdot \Gamma (x,x', \xi , 1)-\epsilon [\Gamma (x,x', \xi , 1)]^2 } a^{\kappa }_{\mu _{\omega }}(x-x',\xi ) dx' d \xi \nonumber \\&= (2 \pi )^{-n} \lim _{\epsilon \rightarrow 0^+} \int \int u(x')e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2}a^{\kappa }_{\mu _{\omega }}(x-x',\xi ) e^{ -\epsilon (\xi +i\kappa \mu _{\omega }(\xi )(x-x'))^2} dx' d \xi . \end{aligned}$$

(5.4)

Now, we will show that we can use the dominated convergence theorem in the right-hand side of (5.4).

The goal is to bound the integral in $x'$ uniformly in $\epsilon$ by an $L^1$ function in $\xi$. Let $Q_{\kappa }(x,x',\xi )\doteq -\epsilon (\xi +i\kappa \mu _{\omega }(\xi )(x-x'))^2$ and note that it follows from (2.6) that there exist constants ${\tilde{A}},{a}>0$ so that

$$\begin{aligned} \begin{array}{c} \mathcal {R} \{Q_{\kappa }(x,x',\xi )\} = -\epsilon (\xi ^2-[\kappa \mu _{\omega }(\xi )]^2\Vert x-x'\Vert ^2) \le -\epsilon a \Vert \xi \Vert ^2,\\ \text{ for } \ x' \in B(0,R), \ x \in B(0,r), \ \Vert \xi \Vert >{\tilde{A}}. \end{array} \end{aligned}$$

(5.5)

Therefore, the trivial bound is not good enough for our purpose. The trick here is to use integration by parts and to do so fix $\xi \in {\mathbb {R}}^n$ and assume without loss of generality that $|\xi _1|=\max \{|\xi _k|: k \in \{1, \dots , n\}\}.$ Then it follows that

$$\begin{aligned} \Vert \xi \Vert = 2 \left( \Vert \xi \Vert - \frac{\Vert \xi \Vert }{2} \right) \le 2 \left( \sqrt{n}|\xi _1| - \frac{\Vert \xi \Vert }{2} \frac{|x_1-x'_1|}{R+r} \right) \end{aligned}$$

(5.6)

for $x' \in B(0,R)$ and $x \in B(0,r)$. Moreover, using (2.6) and (2.12) and increasing ${\tilde{A}}>0$ (if necessary) we have

$$\begin{aligned} \frac{\Vert \xi \Vert }{2} \frac{1}{R+r}&\ge {\kappa 2\sqrt{n}}{D} \, \omega \big ((\Vert \xi \Vert ^2+\delta )^{1/2} \big )\ge \kappa 2\sqrt{n} \, |\mu _\omega (\xi )|, \qquad \Vert \xi \Vert \ge {{\tilde{A}}} \end{aligned}$$

(5.7)

where D is the constant appearing in (2.12). Thus one can use (5.7) to continue estimating (5.6) as

$$\begin{aligned} \begin{array}{c} \Vert \xi \Vert \le 2 \left( \sqrt{n} |i\xi _1| - \kappa 2 \sqrt{n} | \mu _{\omega }(\xi )| |x'_1-x_1| \right) \le 2 \sqrt{n} \big |-i\xi _1 + 2 (x_1-x'_1)\kappa \mu _{\omega }(\xi ) \big |,\\ \text{ for }\ x' \in B(0,R), \ x \in B(0,r) \ \text { and} \ \Vert \xi \Vert >{\tilde{A}} \ (\text{ increasing } {\ \tilde{A}},\ \text{ if } \text{ necessary}). \end{array}\nonumber \\ \end{aligned}$$

(5.8)

Thus, integrating by parts the integral in $x'$ in the right-hand side of (5.4) multiplied by $\Vert \xi \Vert ^{n+1}$, we have

$$\begin{aligned} \Vert \xi \Vert ^{n+1}\bigg |&\int _{{\mathbb {R}}^n} u(x')e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2 }a^\kappa _{ \mu _{\omega }}(x-x',\xi ) e^{-\epsilon (\xi +i\kappa \mu _{\omega }(\xi )(x-x'))^2} dx' \bigg |\nonumber \\&\le \Vert \xi \Vert ^n (4 n)^{\frac{1}{2}} \bigg | \int _{{\mathbb {R}}^n} \!\!\! \partial _{x'_1} \left\{ e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2} \right\} u(x')a^\kappa _{\mu _{\omega }}(x-x',\xi ) e^{-\epsilon (\xi +i\kappa \mu _{\omega }(\xi )(x-x'))^2} dx' \bigg |\nonumber \\&\le (4 n)^{\frac{n+1}{2}} \bigg | \int _{{\mathbb {R}}^n}\!\! e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2} \partial _{x'_1}^{n+1} \left\{ u(x')a^\kappa _{\mu _{\omega }}(x-x',\xi ) e^{-\epsilon (\xi +i\kappa \mu _{\omega }(\xi )(x-x'))^2} \right\} dx' \bigg |, \end{aligned}$$

(5.9)

for $x' \in B(0,R)$, $x \in B(0,r)$ and ${\Vert \xi \Vert }>{\tilde{A}}.$

Moreover, for each $q \in \{1,\dots , n+1\}$, using Faà di Bruno’s formula (see [14, Corollary 2.11])

$$\begin{aligned} \partial _{x'_1}^{q}\left\{ e^{Q_{\kappa }(x,x',\xi )} \right\}&= \sum _{r=1}^q e^{Q_\kappa (x,x',\xi )} \sum _{{\mathfrak {p}}(q,r)} q! \prod _{p=1}^q \frac{[\partial _{x'_{1}}^pQ_{\kappa }(x,x',\xi )]^{k_p}}{k_p! [p!]^{k_p}} \end{aligned}$$

and

$$\begin{aligned} {\mathfrak {p}}(q,r)= \left\{ (k_1, \dots , k_q): k_p \ge 0, \ \sum _{p=1}^qk_p=r, \ \sum _{p=1}^q p k_p=q \right\} . \end{aligned}$$

It is easy to see that there exists a constant $c>0$ such that

$|\partial _{x'_1}Q_{\kappa }(x,x',\xi )| \le \epsilon c \Vert \xi \Vert ^2, \quad |\partial _{x'_1}^2Q_{\kappa }(x,x',\xi )| \le \epsilon c \Vert \xi \Vert ^2, \quad \text{ and } \quad \partial _{x'_1}^pQ_{\kappa }(x,x',\xi ) =0 \ (p \ge 3),$

for $x' \in B(0,R)$, $x \in B(0,r)$ and $\Vert \xi \Vert >{\tilde{A}}$.

Consequently, we will be able to consider the sum over a subset of ${\mathfrak {p}}(q,r)$, considering only derivatives of order less than three. Moreover, since $\displaystyle {r!\sum \nolimits _{\mathfrak { p }(q,r)} \prod \nolimits _{p=1}^q \frac{1}{k_p!}=\left( {\begin{array}{c}q-1\\ r-1\end{array}}\right) }$ (see [14, p. 515]), it follows that

$$\begin{aligned} \left| \partial _{x'_1}^{q}\left\{ e^{Q_{\kappa }(x,x',\xi )} \right\} \right|&\le \sum _{r=1}^q e^{-\epsilon a \Vert \xi \Vert ^2} \sum _{{\mathfrak {p}}_2(q,r)} q! \frac{(\epsilon c \Vert \xi \Vert ^2)^{r}}{k_1!k_2! 2^{k_2}}\nonumber \\&\le q! \sum _{r=1}^q r!\sum _{{\mathfrak {p}}_2(q,r)} \frac{(c/a)^{r}}{k_1!k_2! 2^{k_2}}\nonumber \\&\le q! \sum _{r=1}^q (c/a)^{r} r!\sum _{\mathfrak { p }(q,r)} \prod _{p=1}^q \frac{1}{k_p!}\nonumber \\&= q! \sum _{r=1}^q (c/a)^r \left( {\begin{array}{c}q-1\\ r-1\end{array}}\right) \nonumber \\&\le (1+c/a)^q q! \end{aligned}$$

(5.10)

where ${{\mathfrak {p}}_2(q,r)}=\{(k_1,k_2): k_1+k_2=r \,\,\, \text{ and } \,\,\, k_1+2k_2=q\}$, for $x' \in B(0,R)$, $x \in B(0,r)$ and $\Vert \xi \Vert >{\tilde{A}}$. Thus using the Leibniz rule and (5.10) one can bound the term in (5.9) by an uniform constant $C_n>0$ independent of $\epsilon$,

$$\begin{aligned} \begin{array}{c} \Vert \xi \Vert ^{n+1}\bigg | \displaystyle \int _{{\mathbb {R}}^n} u(x')e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2}a_{\mu _{\omega }}(x-x',\xi ) e^{-\epsilon (\xi +it\kappa \mu _{\omega }(\xi )(x-x'))^2} dx' \bigg | \le C_n,\\ \text {for all} x \in B(0,r) \text {and } \Vert \xi \Vert >{\tilde{A}}. \end{array}\nonumber \\ \end{aligned}$$

(5.11)

Since u is compactly supported, considering (5.4) and taking into account (5.11) we can use the dominated convergence theorem in the right-hand side of (5.4) to conclude that

$$\begin{aligned} u(x)=(2\pi )^{-n} \int _{{\mathbb {R}}^n} {{\mathcal {F}}}_{\mu _{\omega }}^{\kappa } u (x,\xi ) d\xi , \Vert x\Vert <r \end{aligned}$$

for a arbitrary $r>0$ as we wished to prove. $\square$

Lemma 5.2

If $\kappa >0$ and $u \in \mathcal {D}^{\omega }({\mathbb {R}}^n)$ then, for $x \in {\mathbb {R}}^n:$

$$\begin{aligned} u(x)=(2\pi )^{-n} \lim _{j \rightarrow + \infty } \int e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}_{\mu _{\omega }}^{\kappa } u (x,\xi ) d\xi , \quad \text {in}\quad \mathcal {E}^{\omega }({\mathbb {R}}^n). \end{aligned}$$

Proof

Observe that, from Theorem 4.2 for each compact subset $K \subset \mathbb {R}^n$ there exist an open neighborhood V of K and $a,\lambda ,C>0$ such that

$$\begin{aligned} |\partial _x^{\alpha } \mathcal {F}^{\kappa }_{\mu _{\omega }} u(x,\xi )| \le Ce^{-a \omega (\xi )} e^{\frac{1}{\lambda } \varphi ^*(|\alpha |\lambda )}, \quad x \in {V} \,\, \text{ and } \,\, \xi \in \mathbb {R}^n. \end{aligned}$$

(5.12)

Moreover, using Lemma 5.1 it follows that $u(x)=(2\pi )^{-n} \int _{\mathbb {R}^n} \mathcal {F}^{\kappa }_{\mu _{\omega }} u (x,\xi ) d \xi .$ Thus, from (5.12),

$$\begin{aligned} \bigg | \partial _{x}^{\alpha } \bigg \{ u(x) - (2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}}&\mathcal {F}^{\kappa }_{\mu _{\omega }}u(x,\xi ) d \xi \bigg \} \bigg | = \left| \partial _{x}^{\alpha } \left\{ (2\pi )^{-n} \int \left( 1-e^{-\frac{\Vert \xi \Vert ^2}{j}}\right) \mathcal {F}^{\kappa }_{\mu _{\omega }}u(x,\xi ) d \xi \right\} \right| \\&\le (2\pi )^{-n} \int (1-e^{-\frac{\Vert \xi \Vert ^2}{j}}) Ce^{\frac{1}{\lambda } \varphi ^*(|\alpha |\lambda )} e^{-a \omega (\xi )} d \xi \\&\le (2\pi )^{-n} Ce^{\frac{1}{\lambda } \varphi ^*(|\alpha |\lambda )} \int (1-e^{-\frac{\Vert \xi \Vert ^2}{j}}) e^{-a \omega (\xi )} d \xi , \quad x \in K. \end{aligned}$$

The lemma now follows from (2.9) and the dominated convergence theorem. $\square$

5.2 FBI Inversion Formulas for Ultradistributions

The aim of this section is to present two inversion formulas for ultradistributions.

Lemma 5.3

Let $\kappa >0$ be arbitrary. If $u \in {{{\mathcal {D}}}^{\omega }}'({\mathbb {R}}^n)$ and $\psi \in {{{\mathcal {D}}}^{\omega }}({\mathbb {R}}^n)$ then

$$\begin{aligned} \langle u_{j}; \phi \rangle \rightarrow \langle u; \psi \phi \rangle , \quad j \rightarrow + \infty \quad \text{ for } \text{ each } \quad \phi \in \mathcal {D}^{\omega }({\mathbb {R}}^n) \end{aligned}$$

where $u_{j}(x) \doteq (2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}_{\mu _{\omega }}^{\kappa } (\psi u) (x,\xi ) d\xi .$

Remark 5.4

Note that from Lemma 3.5 and Remark 2.11 it follows that $u_j$ is well defined.

Proof

For each $\phi \in \mathcal {D}^{\omega }({\mathbb {R}}^n)$ using Lemma 5.2 and the notation ${\check{\phi }}(x)=\phi (-x)$ we have

$$\begin{aligned} \langle u_{j}, \phi \rangle =&(2 \pi )^{-n} \int \int e^{-\frac{\Vert \xi \Vert ^2}{j}} \left\langle u_{x'}, \psi (x') e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2}a_{\mu _{\omega }}(x-x',\xi ) \right\rangle d\xi \phi (x) dx\\ =&\left\langle u_{x'}, \psi (x')(2 \pi )^{-n} \int \int e^{-\frac{\Vert \xi \Vert ^2}{j}} e^{i(x-x')\xi -\kappa \mu _{\omega }(\xi )(x-x')^2}a_{\mu _{\omega }}(x-x',\xi ) d\xi \phi (x) dx \right\rangle \\ =&\left\langle u_{x'},\psi (x')(2 \pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}_{\mu _{\omega }}^{\kappa } {\check{\phi }}(-x',\xi )d\xi \right\rangle \\ \rightarrow&\,\, \langle u, \psi \phi \rangle , \quad \text{ as } \quad {j \rightarrow +\infty } . \end{aligned}$$

Where in the second equality we can apply similar arguments as in the proof of Lemma 3.4 to obtain the convergence of the Riemann integral in $\mathcal {D}^{\omega }-$topology.

Also, in the second and third equality, it was used that $e^{-\frac{{\Vert \cdot \Vert }^2}{j}} \in L^1 (\mathbb {R}^n)$ and that the support of $\psi$ and $\phi$ are compact subsets of ${\mathbb {R}}^n$. $\square$

Next we will use Lemma 5.3 to obtain another inversion formula which will be used in the proof of Theorem 5.6.

Lemma 5.5

Let $\kappa >0$ be arbitrary. If $u \in {{{\mathcal {D}}}^{\omega }}'({\mathbb {R}}^n)$ , $\psi \in {{{\mathcal {D}}}^{\omega }}({\mathbb {R}}^n)$ then

$$\begin{aligned} \langle {\tilde{u}}_{j}; \phi \rangle \rightarrow \langle u \psi ; \phi \rangle , \quad j \rightarrow + \infty \quad \text{ for } \text{ each } \,\, \phi \in \mathcal {D}^{\omega }({\mathbb {R}}^n); \end{aligned}$$

where ${\tilde{u}}_{j}(x) \doteq (2\pi ^3)^{-n/2} \int \int e^{i\xi (x-t)-\kappa \mu _{\omega } (\xi )(x-t)^2} e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}^{\kappa }_{\mu _{\omega }} (u\psi ) (t,\xi ) \,\,\big (\kappa \mu _{\omega }(\xi )\big )^{n/2}dt d\xi .$

Proof

Since, $e^{-\mu _{\omega }(\xi )(x-\cdot )^2} \in L^1(\mathbb {R}^n)$ (for each fixed $x,\xi \in \mathbb {R}^n$) we can rewrite ${\tilde{u}}_{j}(x)$ as

$$\begin{aligned} {\tilde{u}}_{j}(x)&= (2\pi ^3)^{-\frac{n}{2}} \!\! \int \!\!\!\! \int \!\! e^{-\kappa \mu _{\omega } (\xi )(x-t)^2} \! e^{-\frac{\Vert \xi \Vert ^2}{j}} \!\! \left\langle u, \psi (\cdot ) e^{i(x-\cdot )\xi -\kappa \mu _{\omega }(\xi )(t-\cdot )^2}a^\kappa _{\mu _{\omega }}(t-\cdot ,\xi ) \right\rangle \, \big (\kappa \mu _{\omega }(\xi )\big )^{\frac{n}{2}}dt d\xi \\&= (2\pi ^3)^{-\frac{n}{2}} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} \left\langle u, \psi (\cdot )\int e^{i(x-\cdot )\xi -\kappa \mu _{\omega }(\xi )[(t-\cdot )^2+(x-t)^2]}a ^\kappa _{\mu _{\omega }}(t-\cdot ,\xi ) dt \right\rangle \big (\kappa \mu _{\omega }(\xi )\big )^{\frac{n}{2}} d \xi , \end{aligned}$$

where it was used that the Riemann sum converges to the Riemann integral in the $\mathcal {D}^{\omega }$-topology with respect to the variable t. Using the equations

$$\begin{aligned} 2\left[ t-\frac{1}{2}(x+x')\right] ^2= \frac{1}{2} [(t-x)+(t-x')]^2=\frac{1}{2}(t-x)^2+(t-x)(t-x')+\frac{1}{2}(t-x')^2 \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}(x-x')^2=\frac{1}{2}(x-t)^2+(x-t)(t-x')+\frac{1}{2}(t-x')^2 \end{aligned}$$

it follows that

$$\begin{aligned} {\tilde{u}}_{j}(x)&={ (2\pi ^3)^{\frac{-n}{2}}\!\! \!\int \! \! e^{\frac{-\Vert \xi \Vert ^2}{j}} \!\left\langle u, \psi \!\int \!\! e^{i(x-\cdot )\xi -\kappa \mu _{\omega }(\xi )\left[ \frac{1}{2}(x-\cdot )^2+2(t-\frac{1}{2}(x+\cdot ))^2 \right] }a^\kappa _{\mu _{\omega }}(t-\cdot ,\xi ) dt \right\rangle } \big (\kappa \mu _{\omega }(\xi )\big )^{\frac{n}{2}} d\xi . \end{aligned}$$

(5.13)

Since $e^{-2\kappa \mu _{\omega }(\xi ) (\cdot -\frac{1}{2}(x+x'))^2 } \in L^{1}(\mathbb {R}^n)$ (for each $\xi ,x,x',\kappa$) we consider the change of variables, $-w=t-\frac{1}{2}\left( x+x' \right)$ to write

$$\begin{aligned} \int \!\! e^{-\kappa \mu _{\omega }(\xi )\left\{ 2[t-\frac{1}{2}(x+x')]^2 \right\} }a^\kappa _{\mu _{\omega }}(t-x',\xi ) dt= \int \!\! e^{-\kappa \mu _{\omega } (\xi ) 2w^2}a^\kappa _{\mu _{\omega }}\big ( \tfrac{1}{2}(x-x')-w,\xi \big ) dw. \end{aligned}$$

Moreover, using the identity (see [32, (IX.4.6)])

$$\begin{aligned} \left( \frac{z}{\pi }\right) ^{\frac{1}{2}} \int _{-\infty }^{+\infty } e^{-zy^2}[A+B(\lambda -y)]dy = A+B \lambda , \quad z,A,B,\lambda \in \mathbb {C} \ \text{ and } \ \mathcal {R}\,z >0, \end{aligned}$$

Observe that $a_{\mu _{\omega }}^{\kappa } \big (z,\xi \big )$ is a polynomial of degree n with respect to $z\in \mathbb {R}^n$ (see (1.3)). Additionally, it is a polynomial of degree 1 as a function of $z_j$ for each $j\in \{1,\dots ,n\}$, depending on the parameters $z_k$ for $k\ne j$. Since $\kappa >0$ and $\mu _{\omega }(y) >0$ we obtain

$$\begin{aligned} \left( \frac{2\kappa \mu _{\omega }(\xi )}{\pi } \right) ^{\frac{n}{2}}\int \!\! e^{-\kappa \mu _{\omega } (\xi ) 2w^2}a^\kappa _{\mu _{\omega }}\big ( \tfrac{1}{2}(x-x')-w,\xi \big ) dw = a_{\mu _{\omega }}^{\kappa } \big ( \tfrac{1}{2}(x-x'),\xi \big ). \end{aligned}$$

This together with (5.13) allow us to rewrite ${\tilde{u}}_{j}(x)$ as

$$\begin{aligned} {\tilde{u}}_{j}(x)&=(2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} \left\langle u_{x'},{\psi (x')} e^{i(x-x')\xi -\frac{\kappa }{2}\mu _{\omega }(\xi ) (x-x')^2} a^{\kappa }_{\mu _{\omega }}\big (\tfrac{x}{2}-\tfrac{x'}{2},\xi \big ) \right\rangle d\xi . \nonumber \\&=(2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} \left\langle u_{x'}, \psi (x') e^{i(x-x')\xi -\frac{\kappa }{2}{\mu }(\xi ) (x-x')^2} a_{\mu }^{\kappa /2}\left( x-x',\xi \right) \right\rangle d\xi \nonumber \\&=(2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}_{\mu _{\omega }}^{\kappa /2} (\psi u) (x,\xi ) d\xi . \end{aligned}$$

(5.14)

Where in the second equality we used that $a^{\kappa }_{\mu _{\omega }}\left( \frac{x}{2},\xi \right) =a^{{\kappa }/{2}}_{\mu _{\omega }}\left( x,\xi \right)$. Therefore the result follows from Lemma 5.3. $\square$

5.3 Sufficient Condition of Theorem 1.1

In this section we will use the inversion formula presented in the previous section to prove that a certain decay of the FBI transform in all directions implies $\mathcal {E}^{\omega }$ local regularity.

Theorem 5.6

Let $x_0 \in \mathbb {R}^n,$ $\omega$ and $\sigma$ be weight functions such that $\omega (t)=O(\sigma (t))$ (for $t \rightarrow + \infty$), and $u \in {\mathcal {D}^{\sigma }}' (\mathbb {R}^n)$. If there exist $\psi \in \mathcal {D}^{\sigma }(\mathbb {R}^n)$ and $C,c,\kappa ,r>0$ such that

$$\begin{aligned} |\mathcal {F}^{\kappa }_{\mu _{\sigma }}( \psi u) (t,\xi ) | \le C e^{-c \omega (\xi )}, \quad (t,\xi ) \in B(x_0,r) \times \mathbb {R}^n \end{aligned}$$

(5.15)

then $u \in \mathcal {E}^{\omega }$ in a neighborhood of $x_0$.

Proof

We first note that it follows from Remark 2.11 and inequality (2.12) that there exists $A_1>1$ such that $|\sigma (( \Vert \xi \Vert ^2+\delta _\sigma )^{1/2})| \le e^{-c}\Vert \xi \Vert$ and $|\mu _{\sigma }(\xi )|\le \Vert \xi \Vert ,$ for ${\Vert \xi \Vert }>A_1.$ Moreover, using (2.7) we obtain

$$\begin{aligned} \displaystyle \int _{\Vert \xi \Vert>A_1} e^{-c\omega (\xi )}d\xi \le e^{c\varphi ^*\left( \frac{n+1}{c} \right) }\int _{\Vert \xi \Vert >A_1} \frac{1}{\Vert \xi \Vert ^{n+1}} d\xi <+ \infty . \end{aligned}$$

Moving on we want to apply Lemma 5.5 and in order to do so we write

$$\begin{aligned} \Psi (x,t,\xi ,j)=(2\pi ^3)^{-n/2}e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u)(t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{n/2}, \end{aligned}$$

and

$$\begin{aligned} \int \int \Psi (x,t,\xi ,j)dt d\xi = \int _{U_1}\Psi (x,t,\xi ,j)dtd\xi +\int _{U_2}\Psi (x,t,\xi ,j)dtd\xi + \int _{U_3}\Psi (x,t,\xi ,j)dtd\xi \end{aligned}$$

where, $U_1=\{(t,\xi ): \, \Vert t-x_0\Vert <r, \,\, \Vert \xi \Vert >A_1 \},$ $U_2=\{(t,\xi ): \, \Vert t-x_0\Vert <r, \,\, \Vert \xi \Vert \le A_1 \},$ $U_3=\{(t,\xi ): \, \Vert t-x_0\Vert \ge r, \,\, \xi \in \mathbb {R}^n\}$. Next we will prove that for each $\ell \in \{1,2,3\}$ there exists a function $I_\ell \in \mathcal {E}^{\omega }$ in a neighborhood $U_0$ of $x_0$ such that $\displaystyle \lim _{j \rightarrow + \infty }\int _{U_\ell }\Psi (x,t,\xi ,j)dtd\xi = I_\ell (x)$ in $\mathcal {E}^{\omega }(U_0)$. Therefore, using Lemma 5.5 we will obtain $u=I_1+I_2+I_3$ in a neighborhood of $x_0,$ concluding the proof.

Observe that $\int _{\Vert t-x_0\Vert<r} \int _{\Vert \xi \Vert >A_1} e^{-c\omega (\xi )} d\xi dt <+ \infty .$ Thus, we can use the dominated convergence theorem to conclude that

$$\begin{aligned} \int _{U_1}\Psi (x,t,\xi ,j)dtd\xi \rightarrow I_1(x)\doteq \int _{U_1} (2\pi ^3)^{-n/2}e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u) (t,\xi ) \, [\kappa \mu _{\sigma }(\xi )]^{n/2}dtd\xi , \end{aligned}$$

as $j\rightarrow \infty .$ Moreover, using (3.2) and (5.15) there exists $D>0$ such that, for every $x \in \mathbb {R}^n$ and every $\alpha \in \mathbb {N}_0^n$ it holds

$$\begin{aligned} |\partial _x^{\alpha }I_1(x) |&=\left| \int _{U_1} (2\pi ^3)^{-n/2} \partial _x^{\alpha } \left\{ e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} \right\} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u) (t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{n/2}dtd\xi \right| \\&\le \int _{U_1} \!\! (2\pi ^3)^{-\frac{n}{2}} \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \Vert \xi \Vert ^{|\alpha -\beta |} D^{|\beta |} \beta !^{\frac{1}{2}} [\sigma (({\Vert \xi \Vert ^2+\delta _{\sigma }})^{\frac{1}{2}})]^{\frac{|\beta |}{2}} Ce^{-c\omega (\xi )} [\kappa \mu _{\sigma }(\xi )]^{\frac{n}{2}} dtd\xi . \end{aligned}$$

Since $|\sigma (({ \Vert \xi \Vert +\delta _\sigma })^{1/2})| \le e^{-c}\Vert \xi \Vert$ and $|\mu _{\sigma }(\xi )|\le \Vert \xi \Vert ,$ for $\Vert \xi \Vert >A_1,$ using (2.3), (2.7) and (2.8) there exists $D_0>0$ such that

$$\begin{aligned} D^{|\beta |} |\beta |!^{\frac{1}{2}} [\sigma (({\Vert \xi \Vert +\delta _{\sigma }})^{1/2})]^{\frac{|\beta |}{2}} e^{-c\omega (\xi )}|\kappa \mu _{\sigma }(\xi )|^{\frac{n}{2}}&\le D^{|\beta |} \kappa ^{\frac{n}{2}} |\beta |!^{\frac{1}{2}} e^{-|\beta |c/2}\Vert \xi \Vert ^{\frac{|\beta |+n}{2}} e^{-c\omega (\xi )} \nonumber \\&\le D^{|\beta |} \kappa ^{\frac{n}{2}} (|\beta |+n)!^{\frac{1}{2}}e^{-|\beta |c/2} e^{\frac{c}{2}\varphi ^*(\frac{|\beta |+n}{c})} e^{-\frac{c}{2}\omega (\xi )}\nonumber \\&\le D_0^{|\beta |+n} e^{{c}\varphi ^*(\frac{|\beta |+n}{c})} e^{-\frac{c}{2}\omega (\xi )}\nonumber \\&\le D_0^{|\beta |+n} e^{(c/2)\varphi ^*(\frac{n}{c/2})} e^{(c/2)\varphi ^*(\frac{|\beta |}{c/2})} e^{-\frac{c}{2}\omega (\xi )} . \end{aligned}$$

(5.16)

Thus, using (2.7) there exists $C_1,D_1>0$ such that

$$\begin{aligned} \begin{array}{c} \hspace{-1cm}|\partial _x^{\alpha }I_1(x) |\le C_1 \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) D_1^{|\beta |} \int _{\Vert \xi \Vert>A_1} \Vert \xi \Vert ^{|\alpha -\beta |} e^{(c/2)\varphi ^*\left( \frac{|\beta |}{c/2}\right) } e^{-\frac{c}{2}\omega (\xi )} d\xi \\ \hspace{1.7cm}\le C_1 \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) D_1^{|\beta |} \int _{\Vert \xi \Vert >A_1} e^{\frac{c}{4} \varphi ^*\left( \frac{4}{c} |\alpha -\beta |\right) } e^{\frac{c}{2} \varphi ^*\left( \frac{2}{c} |\beta |\right) } e^{-\frac{c}{4} \omega (\xi )} d\xi ,\\ \text{ for } \text{ every } \ x \in \mathbb {R}^n \ \text{ and } \text{ every } \ \alpha \in \mathbb {N}_0^n. \end{array} \end{aligned}$$

Since $t \mapsto \frac{\varphi ^*(t)}{t}$ is increasing, using (2.3), the last inequality can be further estimated as follows

$$\begin{aligned} |\partial _x^{\alpha }I_1(x) |\le&C_1 \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \int _{\Vert \xi \Vert>A_1} D_1^{|\beta |} e^{\frac{c}{4} \varphi ^*\left( \frac{4}{c} |\alpha -\beta |\right) } e^{\frac{c}{4} \varphi ^*\left( \frac{4}{c} |\beta |\right) } e^{-\frac{c}{4} \omega (\xi )} d\xi \\ \le&C_1 (D_1+1)^{n|\alpha |} e^{\frac{c}{4} \varphi ^*\left( \frac{4}{c} |\alpha |\right) } \int _{\Vert \xi \Vert >A_1} e^{-\frac{c}{4} \omega (\xi )} d\xi ,\\ {}&\quad \text{ for } \text{ every } \ x \in \mathbb {R}^n \ \text{ and } \text{ every } \ \alpha \in \mathbb {N}_0^n. \end{aligned}$$

Now we invoke (2.5) to obtain $D_2>0$ and $\lambda _1>0$ such that $|\partial _x^{\alpha }I_1(x) | \le D_2 e^{\frac{1}{\lambda _1} \varphi ^*(\lambda _1 |\alpha |)},$ for each $(x,\alpha )\in \mathbb {R}^n \times \mathbb {N}_0^n.$ Thus $I_1 \in \mathcal {E}^{\omega }({\mathbb {R}}^n).$ Moreover, reasoning analogously we see that,

$$\begin{aligned} \left| \partial ^{\alpha }_{x} \left\{ \int _{U_1}\Psi (x,t,\xi ,j)dtd\xi - I_1(x)\right\} \right| \le C_1 (D_1+1)^{n|\alpha |} e^{\frac{c}{4} \varphi ^*\left( \frac{4}{c} |\alpha |\right) } \int _{\Vert \xi \Vert >A_1} e^{-\frac{c}{4} \omega (\xi )} \left| e^{-\frac{\Vert \xi \Vert ^2}{j}} -1\right| d\xi . \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{U_1}\Psi (\cdot ,t,\xi ,j)dtd\xi \rightarrow I_1, \quad \text{ as } j\rightarrow + \infty , \quad \text{ in }\,\, \mathcal {E}^{\omega }({\mathbb {R}}^n). \end{aligned}$$

(5.17)

Next we will study $U_2.$ Since $U_2$ is bounded for each $x \in \mathbb {R}^n$ it follows that

$$\begin{aligned} \int _{U_2}\Psi (x,t,\xi ,j)dtd\xi \rightarrow I_2(x)\doteq \int _{U_2} (2\pi ^3)^{-\frac{n}{2}}e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u) (t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{\frac{n}{2}}dtd\xi . \end{aligned}$$

Also using (3.2) and (5.15) there exist constants $C,D>0$ such that for every $\alpha \in \mathbb {N}_0^n$ we have

$$\begin{aligned} |\partial _x^{\alpha }I_2 (x)|&=\left| \int _{U_2} (2\pi ^3)^{-n/2} \partial _x^{\alpha } \left\{ e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} \right\} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u) (t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{n/2}dtd\xi \right| \\&\!\le \int _{U_2} \!\!\!(2\pi ^3)^{-\frac{n}{2}} Ce^{-c\omega (\xi )} [\mu _{\sigma }(\xi )]^{\frac{n}{2}} \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \Vert \xi \Vert ^{|\alpha -\beta |} D^{|\beta |} \beta !^{\frac{1}{2}} [\sigma (({\Vert \xi \Vert ^2+\delta _{\sigma }})^{\frac{1}{2}})]^{\frac{|\beta |}{2}} dtd\xi \\&\!\le \int _{U_2}\!\! (2\pi ^3)^{-n/2} C [\sup _{\Vert \xi \Vert \le A_1}|\mu _{\sigma } (\xi )|]^{n/2} \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) A_1^{|\alpha -\beta |} D^{|\beta |} \beta !^{1/2} [\sigma (({A_1^2+\delta _{\sigma }})^{1/2})]^{|\beta |/2} dtd\xi {.} \end{aligned}$$

Hence, there exist $D_3,D_4>0$ such that $\Vert \partial _x^{\alpha }I_2 \Vert \le D_3 D_4^{|\alpha |} |\alpha |!$, for every $\alpha \in \mathbb {N}_0^n.$ Therefore, using Remark 2.12 it follows that $I_2 \in \mathcal {E}^{\omega }({\mathbb {R}}^n)$. Moreover, one can see that

$$\begin{aligned} \left| \partial ^{\alpha }_{x} \left\{ \int _{U_2}\Psi (x,t,\xi ,j)dtd\xi - I_2(x)\right\} \right| \le D_3D_4^{|\alpha |} |\alpha |! \int _{\Vert \xi \Vert \le A_1} \left| e^{-\frac{\Vert \xi \Vert ^2}{j}} -1\right| d\xi . \end{aligned}$$

Thus,

$$\begin{aligned} \int _{U_2}\Psi (\cdot ,t,\xi ,j)dtd\xi \rightarrow I_2, \quad \text{ as } j\rightarrow +\infty \quad \text{ in }\,\, \mathcal {E}^{\omega }(\mathbb R^n). \end{aligned}$$

(5.18)

Next, in order to study the integral in the region $U_3$, observe that

$$\begin{aligned} \Vert x-t \Vert \ge \Vert t-x_0\Vert -\Vert x-x_0 \Vert \ge r-\frac{r}{2}=\frac{r}{2} \end{aligned}$$

for $\Vert t-x_0\Vert \ge r$ and $\Vert x-x_0\Vert <\frac{r}{2}$.

Thus, using (2.11), (2.12) and Lemma 3.5 there exists $D>0$ such that,

$$\begin{aligned} \left| e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} e^{-\frac{\Vert \xi \Vert ^2}{j}} {{\mathcal {F}}}_{\mu _{\sigma }} ^{{\kappa }}({\psi } u)(t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{\frac{n}{2}} \right|&\le D e^{- \frac{\kappa }{2} \sigma (\delta ^{\frac{1}{2}})(x-t)^2}e^{-\frac{\kappa }{2}\tilde{\sigma } (\xi )\frac{r^2}{4}}e^{\frac{\kappa r^2}{16} \tilde{\sigma } (\xi )}|\tilde{\sigma }(\xi )|^{\frac{n}{2}}\\&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{ = D \left[ \frac{ (\kappa r^2 \tilde{\sigma }(\xi ))^{n}}{16^n n!} \frac{ n!16^n}{(\kappa r^2)^n} \right] ^{\frac{1}{2}} e^{-\frac{\kappa r^2}{16} \tilde{\sigma } (\xi )}e^{- \frac{\kappa }{2} \sigma (\delta ^{\frac{1}{2}})(x-t)^2}}\\&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\le \frac{n!^{\frac{1}{2}}16^{\frac{n}{2}}}{\kappa ^{\frac{n}{2}} r^n} D e^{-\kappa \tilde{\sigma } (\xi )\frac{r^2}{32}} e^{-\kappa \sigma (\delta ^{\frac{1}{2}})(x-t)^2} \end{aligned}$$

for $\tilde{\sigma }(\xi )=\sigma ( (\delta _{\sigma }+\Vert \xi \Vert ^2)^{1/2})$, $(t,\xi ) \in U_3$ and $\Vert x-x_0\Vert <\frac{r}{2},$ where it was used that $\left[ \frac{(\kappa r^2 \tilde{\sigma }(\xi ))^n}{16^n n!} \right] ^{\frac{1}{2}} \le \left[ e^{\frac{\kappa r^2 \tilde{\sigma }(\xi )}{16}} \right] ^{\frac{1}{2}}$. Hence, using the dominated convergence theorem (recalling that $\int e^{-\kappa \frac{r^2}{16} \tilde{\sigma }(\xi )}d\xi < + \infty$ and $\int e^{-\kappa \sigma (\delta ^{1/2})(x-t)^2} dt < + \infty$),

$$\begin{aligned}&\displaystyle \int _{U_3}\Psi (x,t,\xi ,j)dtd\xi \rightarrow I_3(x) \nonumber \\&\quad \displaystyle =\int _{U_3} (2\pi ^3)^{-n/2}e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u) (t,\xi ) \,\,\, [\mu _{\sigma }(\xi )]^{n/2}dtd\xi ,\nonumber \\ {}&\qquad \quad \text{ whenever } \ \Vert x-x_0\Vert <\frac{r}{2}. \end{aligned}$$

(5.19)

Moreover, using the Leibniz rule, Remark 3.3, Lemma 3.5, (2.3), (2.7) and (2.8), for each $\theta ,\lambda >0$ to be chosen there exist $D_3,D_4,D_5>0$ such that

$$\begin{aligned}&\Big |\partial ^{\alpha }_x \Big \{ e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} {{\mathcal {F}}}_{\mu _{\sigma }}^\kappa (\psi u)(t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{\frac{n}{2}} \Big \} \Big |\nonumber \\ {}&\quad \le \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \Vert \xi \Vert ^{|\alpha -\beta |} D_3^{|\beta |} \beta ! e^{-\frac{\kappa }{2} \sigma (({\Vert \xi \Vert ^2+\delta _{\sigma }})^{1/2})\Vert x-t\Vert ^2} e^{\theta \sigma (({\Vert \xi \Vert ^2+\delta _{\sigma }})^{1/2})} [\mu _{\sigma }(\xi )]^{\frac{n}{2}}\nonumber \\ {}&\quad \le \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) e^{\frac{1}{\lambda } \varphi ^*(\lambda |\alpha |)} e^{\frac{1}{\lambda }\omega (\xi )} D_4^{|\beta |} [\mu _{\sigma }(\xi )]^{\frac{n}{2}} e^{-\frac{\kappa r^2 }{16} \sigma (({\Vert \xi \Vert ^2+\delta _{\sigma }})^{1/2})}\nonumber \\&\qquad \quad e^{-\frac{\kappa }{16} \sigma (\sqrt{\delta _{\sigma }})\Vert x-t\Vert ^2} e^{\theta \sigma (({\Vert \xi \Vert ^2+\delta _\sigma })^{1/2})}\nonumber \\ {}&\quad \le D_5^{|\alpha |} e^{\frac{\varphi ^*(\lambda |\alpha |)}{\lambda } } e^{\frac{\omega (\xi )}{\lambda }} e^{-\frac{\kappa r^2 }{32} \sigma (({\Vert \xi \Vert ^2+\delta _{\sigma }})^{1/2})} e^{- \kappa _{\sigma } \Vert x-t\Vert ^2} [\mu _{\sigma }(\xi )]^{\frac{n}{2}},\nonumber \\ {}&\qquad \text{ for } \ \Vert t-x_0\Vert \ge r \ \text{ and } \ \Vert x-x_0 \Vert <\frac{r}{2} \end{aligned}$$

where in the last inequality we fixed $\theta \doteq \frac{\kappa r^2 }{ 32}$ and $\kappa _{\sigma } \doteq \frac{\kappa }{16} \sigma (\sqrt{\delta })$. Moving on using (2.5) we see that there exist $D_6>0$ and $\lambda _*>0$ such that the last inequality can be further estimated as

$$\begin{aligned}&\Big |\partial ^{\alpha }_x \Big \{ e^{i\xi (x-t)-\kappa \mu _{\sigma } (\xi )(x-t)^2} {{\mathcal {F}}}_{\mu _{\sigma }}^{\kappa } (\psi u)(t,\xi ) \,\,\, [\kappa \mu _{\sigma }(\xi )]^{\frac{n}{2}} \Big \} \Big |\nonumber \\&\quad \le {D_6}\,e^{\frac{\varphi ^*(\lambda _* |\alpha |)}{\lambda _*} } e^{\frac{\omega (\xi )}{\lambda }} e^{-\frac{\kappa r^2}{32} \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} e^{- \kappa _{\sigma }\Vert x-t\Vert ^2} |\mu _{\sigma }(\xi )|^{\frac{n}{2}}\nonumber \\&\qquad \text{ for } \ \Vert t-x_0\Vert \ge r \ \text{ and } \ \Vert x-x_0 \Vert <\frac{r}{2}. \end{aligned}$$

(5.20)

Moreover, from the fact that

$$\begin{aligned} \int _{\Vert t-x_0\Vert >r} e^{- \kappa _{\sigma }\Vert x-t\Vert ^2} dt \le \int _{\mathbb {R}^n} e^{- \kappa _{\sigma } y^2} dy <+ \infty \end{aligned}$$

and considering $A_3,c_1>0$ so that $\omega (\xi ) \le c_1 \sigma (\xi ),$ for $\Vert \xi \Vert >A_3,$ it follows that

$$\begin{aligned} \int e^{\frac{\omega (\xi )}{\lambda }} e^{-\frac{\kappa r^2{n}}{32} \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} | \mu _{\sigma }(\xi )|^{\frac{n}{2}} d\xi \le&\int _{\Vert \xi \Vert \le A_3} e^{\frac{\omega (\xi )}{\lambda }} e^{-\frac{\kappa r^2}{32} \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} |\mu _{\sigma }(\xi )|^{\frac{n}{2}} d\xi \\&+\int _{\Vert \xi \Vert > A_3} e^{\frac{c_1\sigma (({\Vert \xi \Vert ^2 +\delta })^{1/2})}{\lambda }} e^{-\frac{\kappa r^2 }{32} \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} |\mu _{\sigma }(\xi )|^{\frac{n}{2}} d\xi . \end{aligned}$$

Note that, the first integral is finite and choosing $\lambda \doteq \frac{{64}\, c_1}{\kappa r^2 }$ it follows from (2.12), (2.7) and Remark 2.11, that

$$\begin{aligned}&\int _{\Vert \xi \Vert> A_3}\!\! e^{\frac{c_1\sigma (({\Vert \xi \Vert ^2 +\delta })^{1/2})}{\lambda }} e^{-\frac{\kappa r^2 }{{32}} \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} | \mu _{\sigma }(\xi )|^{\frac{n}{2}} d\xi \nonumber \\&\qquad = \int _{\Vert \xi \Vert > A_3} \!\! e^{-\frac{\kappa r^2}{{64}} \sigma (({\Vert \xi \Vert ^2+\delta })^{1/2})} |\mu _{\sigma }(\xi )|^{\frac{n}{2}} d\xi < + \infty . \end{aligned}$$

(5.21)

Thus, summing up (5.19), (5.20) and (5.21) we see that there exists $D_7,\lambda _*>0$ such that

$$\begin{aligned} | \partial _x^{\alpha } I_3(x) |&\le D_7e^{\frac{1}{\lambda _*} \varphi ^*(\lambda _*|\alpha |)}, \quad \forall (x,\alpha ) \in B\left( x_0, \tfrac{r}{2}\right) \times \mathbb {N}_0^n. \end{aligned}$$

This shows that $I_3 \in \mathcal {E}^{\omega }.$ In addition to that and similarly as before it follows that

$$\begin{aligned} \left| \partial ^{\alpha } \left\{ \int _{U_3}\Psi (x,t,\xi ,j)dtd\xi - I_3(x)\right\} \right| \le D_7e^{\frac{1}{\lambda _*} \varphi ^*(\lambda _*|\alpha |)} \int _{\Vert \xi \Vert \le A_1} \left| e^{-\frac{\Vert \xi \Vert ^2}{j}} -1\right| d\xi . \end{aligned}$$

Hence

$$\begin{aligned} \int _{U_3}\Psi (\cdot ,t,\xi ,j)dtd\xi \rightarrow I_3, \quad {\text{ as } j\rightarrow + \infty ,} \quad \text{ in }\,\, \mathcal {E}^{\omega }(B(x_0,{\tfrac{r}{2}})). \end{aligned}$$

(5.22)

Therefore, using (5.17), (5.18), (5.22) and Lemma 5.5 it follows that $u=I_1+I_2+I_3$ in $B(x_0,\tfrac{r}{2}),$ which concludes the proof. $\square$

6 A Characterization of Ultradifferentiable Iterates of Constant Coefficients Operators

Let $P(\xi )= \sum _{|\alpha | \le m} a_{\alpha } \xi ^{\alpha }$ be a polynomial function of degree m. This section is dedicated to the characterization of the space $\mathcal {E}^{\omega }(\Omega ;P)$ (see Definition 1.2) using a FBI transform.

Remark 6.1

From now on we will consider $0<\rho \le 1$ such that (1.8) is satisfied and denote $\sigma (t)=\sigma _{\omega ,\rho }(t)=\omega (t^{\rho }).$ It is important to note that, if $\omega$ is a weight function and $0<\rho \le 1$ then $\sigma$ is a weight function.

Theorem 6.2

Let $x_0 \in \mathbb {R}^n,$ $\omega$ be a weight function and P(D) be a constant coefficient hypoelliptic linear operator of order m together with its hypoelliptic index $\rho$ (satisfying (1.8)). Let $\sigma (t)=\omega (t^{\rho })$ and $u \in {\mathcal {D}^{\sigma }}' (\mathbb {R}^n)\subset {\mathcal {D}^{\omega }}' (\mathbb {R}^n)$. The following conditions are equivalent:

• 1. There exists a neighborhood $\Omega$ of $x_0$ such that $u \in \mathcal {E}^{\omega }(\Omega ;P)$.

• 2. There exist $\phi \in \mathcal {D}^{\omega }(\mathbb {R}^n)$ (such that $\phi \equiv 1$ in a neighborhood of $x_ 0$), $C,\lambda , c>0$ and a neighborhood V of $x_0$ such that

  $$\begin{aligned} |\mathcal {F}_{\mu _{\sigma }} \left( \phi \, [P(D)]^{N} ( u)\right) (x,\xi ) | \le C e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )} e^{-c \omega (\Vert \xi \Vert ^{\rho })}, \quad (x,\xi ,N) \in V \times \mathbb {R}^n \times \mathbb {N}_0,\nonumber \\ \end{aligned}$$

  (6.1)

where we are denoting $\varphi ^*(x)=\varphi _{\omega }^*(x) \doteq \sup \{xy-\varphi (y): \, y \ge 0\},$ for each $x>0$ and $\mu _{\sigma }$ is the function obtained from Proposition 2.20 when applied to $\sigma$ instead of $\omega .$

Proof

${ (1) \Rightarrow (2) }$: Let $R\in (0,1/3)$ be such that $B(x_0,3R) \subset \Omega$. Set $0<r<R$ and $\phi \in \mathcal {D}^{\omega }(B(x_0,R))$ such that $\phi \equiv 1$ in $B(x_0,r).$ Also, considering $\psi \in \mathcal {D}^{\omega } (B(x_0,2R))$ such that $\psi \equiv 1$ in $B(x_0,R),$ it follows that

$$\begin{aligned} \mathcal {F}_{\mu _{\sigma }}\left( \phi [P(D)]^{N} ( u)\right) (x,\xi ) =&\int \phi (y) [P(D)]^{N} ( u)(y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\\ =&\int \phi (y) [P(D)]^{N} ( u)(y) \, \psi (y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\\ =&\int [\phi (y)-1] [P(D)]^{N} ( u)(y) \, \psi (y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\\&+ \int [P(D)]^{N} u(y) \, \psi (y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\\ =:&I_1(x,\xi )+I_2(x,\xi ), \end{aligned}$$

where $x,\xi \in \mathbb {R}^n$,

$$\begin{aligned} I_1(x,\xi ) \doteq \int [\phi (y)-1] [P(D)]^{N} ( u)(y) \, \psi (y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy \end{aligned}$$

(6.2)

and

$$\begin{aligned} I_2(x,\xi ) \doteq \int [P(D)]^{N} u(y) \, \psi (y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy. \end{aligned}$$

(6.3)

Next we will study $I_1$ and $I_2.$ First, denoting $K=\text{ supp } \psi$ and using Hölder inequality,

$$\begin{aligned} |I_1(x,\xi )| \le \Vert [P(D)]^{N} u \Vert _{L^2(K)} \left[ \int _{r\le |y-x_0|\le 2R} \left| [\phi (y)-1] \, \psi (y) e^{i(x-y)\xi -\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right| ^2 dy\right] ^{\frac{1}{2}} \end{aligned}$$

Since $u\in \mathcal {E}^{\omega }(\Omega ;P)$ and $K \subset \Omega$, recalling Definition 1.2, there exists $C_1,C>0$ and $\lambda >0$ such that

$$\begin{aligned} |I_1(x,\xi )|&\le C_1e^{\frac{1}{\lambda } \varphi ^*(\lambda m N)} \left[ \int _{r\le |y-x_0|\le 2R} \left| [\phi (y)-1] \, \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right| ^2 dy\right] ^{1/2}\\&\le Ce^{\frac{1}{\lambda } \varphi ^*(\lambda m N)} \left[ \sup _{r\le |y-x_0|\le 2R}e^{-2\sigma (\Vert \xi \Vert )(x-y)^2} \right] ^{1/2} \le Ce^{\frac{1}{\lambda } \varphi ^*(\lambda m N)} e^{-\omega (\Vert \xi \Vert ^{\rho })\frac{r^2}{4}} \end{aligned}$$

for each x satisfying $|x-x_0|\le \frac{r}{2}$ and $\xi \in \mathbb {R}^n;$ where we also use Proposition 2.20.

Next we will consider the term $I_2$ given by (6.3). Let $\ell$ be an arbitrary positive integer. Using (1.8) we obtain for $\Vert \xi \Vert \ge K,$ with K the constant appearing in (1.8), and $u_n:= [P(D)]^Nu$

$$\begin{aligned} |I_2 (x,\xi )|&= \left| \int e^{i(x-y)\xi } u_N(y) \, \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\right| \nonumber \\ {}&\le \frac{1}{(C\Vert \xi \Vert ^{\rho m})^\ell }\left| \int [P(\xi )]^{\ell } e^{i(x-y)\xi } u_N(y) \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\right| \nonumber \\ {}&= \frac{1}{(C\Vert \xi \Vert ^{\rho m})^\ell }\left| \int [P(-D_y)]^{\ell } \left\{ e^{i(x-y)\xi } \right\} u_N(y) \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) dy\right| \nonumber \\&= \frac{1}{(C\Vert \xi \Vert ^{\rho m})^\ell }\left| \int e^{i(x-y)\xi } [P(D_y)]^{\ell } \left\{ u_N(y) \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right\} dy\right| . \end{aligned}$$

(6.4)

In order to study the above integral we first recall that

$$\begin{aligned} P(D)(fg)=\sum _{|\beta |\le m} \frac{1}{\beta !} P^{(\beta )}g\,\times \, D^{\beta } f, \quad \forall f,g \in C^{\infty }, \end{aligned}$$

where we denote $P^{(\beta )}g \doteq (\partial _{\xi }^{\beta }P)(D)g$ for each $\beta \in {\mathbb {N}}^n_0$. Using the linearity of P(D) it follows that

$$\begin{aligned} P(D)\circ P(D)(fg)=\sum _{|\beta |\le m} \frac{1}{\beta !} P(D) \left\{ P^{(\beta )}g\,\times \, D^{\beta } f\right\} , \quad \forall f,g \in C^{\infty }. \end{aligned}$$

Hence,

$$\begin{aligned}{} & {} [P(D)]^{\ell }(fg)=\sum _{|\beta _1|\le m} \frac{1}{\beta _1!} \dots \sum _{|\beta _\ell |\le m} \frac{1}{\beta _\ell !} [ P^{(\beta _1)}\circ \dots \circ P^{(\beta _\ell )}]g\\{} & {} \quad \times \, D^{\beta _1+\dots + \beta _\ell } f, \quad \forall f,g \in C^{\infty }. \end{aligned}$$

Thus, inserting the last expression in (6.4) it follows that,

$$\begin{aligned} |I_2(x,\xi )|&\le \frac{ 1}{(C\Vert \xi \Vert ^{\rho m})^\ell } \! \sum _{|\beta _1|\le m}\! \frac{1}{\beta _1!}\! \dots \!\!\!\sum _{|\beta _\ell |\le m}\! \frac{1}{\beta _\ell !} \int _{B(x_0,2R)}\! \left| [P^{(\beta _1)}\circ \dots \circ P^{(\beta _\ell )}u_N(y)\right| \,\nonumber \\&\quad \times \left| \partial _y^{\beta _1+\dots + \beta _\ell } \left\{ \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} \! a_{\mu _{\sigma }}(x-y,\xi ) \right\} \right| dy\nonumber \\&\le \frac{ 1}{(C\Vert \xi \Vert ^{\rho m})^\ell } \sum _{|\beta _1|\le m} \frac{1}{\beta _1!} \dots \sum _{|\beta _\ell |\le m} \frac{1}{\beta _\ell !} \Vert [P^{(\beta _1)}\circ \dots \circ P^{(\beta _\ell )}]u_N\Vert _{L^2(B(x_0,2R))} \nonumber \\&\quad \times \left\| \partial _{y}^{\beta _1+\dots + \beta _\ell } \left\{ \psi (\cdot ) e^{-\mu _{\sigma }(\xi )(x-\cdot )^2} \! a_{\mu _{\sigma }}(x-\cdot ,\xi ) \right\} \right\| _{L^2(B(x_0,2R))} \end{aligned}$$

(6.5)

where $\Vert \xi \Vert >K$. Observe that, since $\sigma$ is increasing it follows that $\sigma ((\Vert \xi \Vert ^2 +\delta _{\sigma })^{1/2})\le \sigma (\Vert \xi \Vert +\delta _{\sigma }^{{1/2}}),$ for each $\xi \in \mathbb {R}^n$. Thus, using Remark 3.3 for $\sigma$ instead of $\omega$, Leibniz’s rule and the fact that $\psi \in \mathcal {D}^{\omega }$ we obtain that for each $\theta >0$ there exist $\lambda _1, D >0$ such that

$$\begin{aligned} \left| \partial ^{\eta }_y \left\{ \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right\} \right|&\le \sum _{\beta \le \eta } \left( {\begin{array}{c}\eta \\ \beta \end{array}}\right) e^{\frac{1}{\lambda _1} \varphi ^*(\lambda _1 |\eta - \beta |)} {D}^{|\beta |} \beta ! e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) }, \end{aligned}$$

for each $\eta \in \mathbb {N}_0^n.$ Thus, using (2.3), (2.5) and (2.8), there exist $C_2,\lambda _2>0$ such that

$$\begin{aligned} \left| \partial ^{\eta }_y \left\{ \psi (y) e^{-\mu _{\sigma }(\xi )(x-y)^2} a_{\mu _{\sigma }}(x-y,\xi ) \right\} \right|&\le C_2 e^{\frac{1}{\lambda _2} \varphi ^*(\lambda _2 |\eta |)} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) }. \end{aligned}$$

Hence, using (2.3)

$$\begin{aligned} \left\| \partial _y^{\beta _1+\dots + \beta _\ell } \left\{ \psi (\cdot ) e^{-\mu _{\sigma }(\xi )(x-\cdot )^2} \! a_{\mu _{\sigma }}(x-\cdot ,\xi ) \right\} \right\| _{L^2(x_0,2R)} \le C_2e^{\sum _{j=1}^{\ell }\frac{1}{2^j\lambda _2} \varphi ^*(2^j\lambda _2 |\beta _j|)} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) }.\nonumber \\ \end{aligned}$$

(6.6)

Moreover, using Corollary 6.7 there exists $C>1$ such that,

$$\begin{aligned}&\Vert [P^{(\beta _1)}\circ \dots \circ P^{(\beta _\ell )}]u_N\Vert _{L^2(B(x_0,2R))}\\&\quad \le C \epsilon _1^{|\beta _1|} \!\!\sup _{j\in \{0,1\}}\!\!\left\{ \Vert [P^j\!\circ \! P^{(\beta _2)}\!\circ \! \dots \! \circ \! P^{(\beta _\ell )}]u_N\Vert _{L^2(B(x_0,2R+\epsilon _1))} \right\} \\&\quad \le C \epsilon _1^{|\beta _1|} \sup _{j\in \{0,1\}}\left\{ \Vert [P^{(\beta _2)}\circ \dots \circ P^{(\beta _\ell )}\circ P^j]u_N\Vert _{L^2(B(x_0,2R+\epsilon _1))} \right\} \\&\quad \le C^{\ell } \epsilon _1^{|\beta _1|} \dots \epsilon _{\ell }^{|\beta _{\ell }|}\!\!\!\!\! \sup _{j_1,\dots j_\ell \in \{0,1\}} \Vert [P^{j_{\ell }}\circ \dots \circ P^{j_1}]u_N\Vert _{L^2(B(x_0,2R+\epsilon _1+ \dots + \epsilon _\ell ))}\\&\quad \le C^{\ell } \epsilon _1^{|\beta _1|} \dots \epsilon _{\ell }^{|\beta _{\ell }|}\!\!\!\!\! \sup _{j_1,\dots j_\ell \in \{0,1\}} \Vert [P(D)]^{j_{\ell }+ \dots + j_1 +N}u\Vert _{L^2(B(x_0,2R+\epsilon _1+ \dots + \epsilon _\ell ))} \end{aligned}$$

for $\epsilon _j=\left[ \frac{\beta _j!}{e^{ [1/(2^j\lambda _2) ]\varphi ^*(2^j\lambda _2 |\beta _j|)}} \right] ^{1/|\beta _j|} \frac{R}{\ell e^{1/(2^j\lambda _2)} C_{\omega }^{1/(2^j\lambda _2|\beta _j|)} },$ where $j \in \{1, \dots , \ell \}$ and $C_{\omega }>1$ is such that $e^{\omega (t)} \le C_{\omega } e^{t}$ (for each $t\ge 0$). Observe that using (2.7) it follows

$$\begin{aligned} \epsilon _j \le \left[ e^{\frac{1}{2^j\lambda _2 }\omega (|\beta _j|)} \right] ^{\frac{1}{|\beta _j|}} \frac{R}{\ell e^{\frac{1}{2^j\lambda _2}} C_{\omega }^{\frac{1}{2^j\lambda _2|\beta _j|}}} \le \left[ C_{\omega }^{\frac{1}{2^j\lambda _2}} e^{\frac{1}{2^j\lambda _2 }|\beta _j|} \right] ^{\frac{1}{|\beta _j|}} \frac{R}{\ell e^{\frac{1}{2^j\lambda _2}} C_{\omega }^{\frac{1}{2^j\lambda _2|\beta _j|}}}=\frac{R}{\ell }, \end{aligned}$$

where $j\in \{1, \dots , \ell \}.$ Thus, $\frac{R}{\ell e^{1/(2^j\lambda _2)} C_{\omega }^{1/(2^j\lambda _2|\beta _j|)} }<1$ (for $j\in \{1, \dots , \ell \}$). Moreover, since $B(x_0,3R) \subset \Omega$ and $u\in \mathcal {E}^{\omega }(\Omega ;P)$ there exist $C_1,\lambda >0$ such that

$$\begin{aligned} \Vert [P^{(\beta _1)}\!\circ \! \cdots \! \circ \! P^{(\beta _\ell )}]u_N\Vert _{L^2(B(x_0,2R))}&\le C^{\ell } \epsilon _1^{|\beta _1|} \cdots \epsilon _{\ell }^{|\beta _{\ell }|}\!\!\!\!\! \sup _{j_1,\dots j_\ell \in \{0,1\}} \Vert [P(D)]^{j_{\ell }+ \cdots + j_1 +N}u\Vert _{L^2(B(x_0,3R))}\nonumber \\ \!\!&\le C^{\ell } \frac{\beta _1!}{e^{ [1/(2^1\lambda _2)] \varphi ^*(2^1\lambda _2 |\beta _1|)}} \cdots \frac{\beta _\ell !}{e^{ [1/(2^\ell \lambda _2)] \varphi ^*(2^\ell \lambda _2 |\beta _\ell |)}} C_1e^{\frac{1}{\lambda } \varphi ^*(\lambda m (\ell +N))}. \end{aligned}$$

(6.7)

Hence, putting (6.5), (6.6) and (6.7) together we obtain,

$$\begin{aligned} |I_2(x,\xi )|&\le \frac{ 1}{(C\Vert \xi \Vert ^{\rho m})^\ell } \sum _{|\beta _1|\le m} \frac{1}{\beta _1!} \cdots \sum _{|\beta _\ell |\le m} \frac{1}{\beta _\ell !} C_1C^{\ell }\frac{\beta _1!}{e^{ [1/(2^1\lambda _2)] \varphi ^*(2^1\lambda _2 |\beta _1|)}} \cdots \frac{\beta _\ell !}{e^{ [1/(2^\ell \lambda _2)] \varphi ^*(2^\ell \lambda _2 |\beta _\ell |)}} \nonumber \\&\quad \times e^{\frac{1}{\lambda }\varphi ^*(\lambda m (\ell +N))} C_2e^{\sum _{j=1}^{\ell }\frac{1}{2^j\lambda _2} \varphi ^*(2^j\lambda _2 |\beta _j|)} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) } \nonumber \\&= \frac{ 1}{\Vert \xi \Vert ^{\rho m \ell } } \sum _{|\beta _1|\le m} \cdots \sum _{|\beta _\ell |\le m} C_1 e^{\frac{1}{\lambda }\varphi ^*(\lambda m (\ell +N))} C_2 e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) }. \end{aligned}$$

(6.8)

Moreover, using (2.3) and (2.5) we see that denoting $C_4=C_1C_2$ and $C_5=\sum _{|\beta |\le m} 1$ there exist positive constants $C_6,\lambda _3,\lambda _4$ such that,

$$\begin{aligned} |I_2(x,\xi )|\le&\frac{ 1}{\Vert \xi \Vert ^{\rho m\ell } } C_4 C_5^{\ell } e^{\frac{1}{\lambda }\varphi ^*(\lambda m (\ell +N))} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) } \le \frac{ e^{\frac{1}{\lambda _3}\varphi ^*(\lambda _3 m \ell )}}{\Vert \xi \Vert ^{\rho m\ell } } C_6 e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) } \nonumber \\ =&\left( \frac{ e^{\frac{1}{m\lambda _3}\varphi ^*(m\lambda _3 \ell )}}{\Vert \xi \Vert ^{\rho \ell } } \right) ^{m} C_6 e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) }. \end{aligned}$$

(6.9)

Since (6.9) holds true for every $\ell \in {\mathbb {N}}$ one can take the infimum in $\ell$ and use [17, Lemma 1.4] to obtain,

$$\begin{aligned} |I_2(x,\xi )|&\le \left( e^{\log \Vert \xi \Vert ^{\rho }} e^{-\frac{1}{m\lambda _3} \omega (\Vert \xi \Vert ^\rho )}\right) ^{m} C_6 e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)} e^{\theta \sigma (\Vert \xi \Vert +\delta _{\sigma }^{1/2}) }. \end{aligned}$$

Now one can use ($\gamma$) to get $A>0$ such that $\log \Vert \xi \Vert ^{\rho } \le \frac{\omega (\Vert \xi \Vert ^{\rho })}{2\,m\lambda _3},$ when $\Vert \xi \Vert >A.$ Thus $m\log (\Vert \xi \Vert ^{\rho } )-\frac{1}{\lambda _3} \omega (\Vert \xi \Vert ^\rho ) \le -\frac{1}{2\lambda _3} \omega (\Vert \xi \Vert ^\rho ),$ when $\Vert \xi \Vert >A.$ This, together with the fact that $0<\rho \le 1$, that $\omega$ is a increasing and ($\alpha$)implies that $\sigma (\Vert \xi \Vert + \delta _{\sigma }^{1/2}) =\omega ( [\Vert \xi \Vert + \delta _{\sigma }^{1/2}]^{\rho })\le \omega ( \Vert \xi \Vert ^{\rho } + \delta _{\sigma }^{\rho /2}) \le \omega (e\Vert \xi \Vert ^\rho ) \le L[\omega (\Vert \xi \Vert ^{\rho })+1],$ when $\Vert \xi \Vert \ge \frac{\delta _{\sigma }^{1/2}}{(e-1)^{1/\rho }}.$ Hence, choosing $\theta =\frac{1}{4\lambda _3L}$ it follows that

$$\begin{aligned} |I_2(x,\xi )|&\le e^{-\frac{1}{2\lambda _3} \omega (\Vert \xi \Vert ^\rho )} C_6 e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)} e^{\theta L[\omega (\Vert \xi \Vert ^{\rho })+1] } {= C_6 e^{\theta L} e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)} e^{\theta L\omega (\Vert \xi \Vert ^{\rho }) } e^{-\frac{1}{2\lambda _3} \omega (\Vert \xi \Vert ^\rho )}}\\&\le C_6 e^{\theta L} e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)} e^{-\frac{1}{4\lambda _3}\omega (\Vert \xi \Vert ^\rho )}, \end{aligned}$$

for $\Vert \xi \Vert \ge \max \left\{ A;\,\, \frac{\delta }{(e-1)^{1/\rho }} \right\} .$ Therefore, there exists $C_7>0$ such that

$$\begin{aligned}|I_2(x,\xi )|\le C_7 e^{-a \omega (\Vert \xi \Vert ^\rho )} e^{\frac{1}{\lambda _4}\varphi ^*(\lambda _4 m N)}, \quad \forall \xi \in \mathbb {R}^n. \,\,\, \end{aligned}$$

${(2) \Rightarrow (1):}$ Consider $0<r<R$, $\phi \in \mathcal {D}^{\omega }(B(x_0,R))$ (such that $\phi \equiv 1$ in $B(x_ 0,r)$) and $C,\lambda , c>0$ such that

$$\begin{aligned} |\mathcal {F}_{\mu _{\sigma }} \left( \phi [P(D)]^{N} ( u)\right) (x,\xi ) | \le C e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )} e^{-c \omega (\Vert \xi \Vert ^{\rho })}, \quad (x,\xi ,N) \in B(x_0,r) \times \mathbb {R}^n \times \mathbb {N}_0.\nonumber \\ \end{aligned}$$

(6.10)

First, observe that,

from Theorem 1.1 we can conclude that there exists $0<\delta <r$ such that $u \in \mathcal {E}^{\sigma }(B(x_0,\delta )).$ In order to prove that $u \in \mathcal {E}^{\omega }(B(x_0,\delta );P)$ we will consider a compact set $K\subset B(x_0,\delta )$. Since $\phi \equiv 1$ in K it follows that $[P(D)]^N(u)(y)= \phi (y) [P(D)]^N(u)(y),$ for each $y\in K$.

Moreover, given $v \in {{\mathcal {D}}^{\sigma }}' (\mathbb {R}^n)$ and denoting

$$\begin{aligned} v^{\phi }_j(x) = (2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} \mathcal F_{\mu _{\sigma }} (\phi v)(x,\xi ) d\xi \end{aligned}$$

(6.11)

it follows from Lemma 5.3 that $\langle v^{\phi }_j;\psi \rangle \rightarrow \langle v \phi ;\psi \rangle$ (for each $\psi \in {\mathcal {D}^{\sigma }}(\mathbb {R}^n)$).

Furthermore for each $N\in \mathbb {N}_0$ it follows from (6.10) that

$$\begin{aligned}{} & {} (2\pi )^{-n} \int e^{-\frac{\Vert \xi \Vert ^2}{j}} {\mathcal {F}}_{\mu _{\sigma }} (\phi [P(D)]^N(u))(x,\xi ) d\xi \\{} & {} \quad \rightarrow (2\pi )^{-n} \int \mathcal F_{\mu _{\sigma }} ( \phi [P(D)]^N(u))(x,\xi ) d\xi ,\,\, \forall x \in K. \end{aligned}$$

Thus,

$$\begin{aligned} {}[P(D)]^N(u)(x)&=\phi (x) [P(D)]^N(u)(x)\\&= (2\pi )^{-n} \int \mathcal F_{\mu _{\sigma }} ( \phi [P(D)]^N(u))(x,\xi ) d\xi ,\,\, x \in K. \end{aligned}$$

Hence, using (6.10),

$$\begin{aligned} \big |[P(D)]^N(u))(x) \big | \le (2\pi )^{-n} C\int e^{-c \omega (\Vert \xi \Vert ^{\rho })} d\xi \,\, e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )} =C_1 e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )},\quad x \in K \end{aligned}$$

where $C_1=(2\pi )^{-n} C\int e^{-c \omega (\Vert \xi \Vert ^{\rho })} d\xi$. Therefore,

$$\begin{aligned} \big \Vert [P(D)]^N(u)) \big \Vert _{L^2(K)}\le C_1 \int _{K} 1 \,dx \ e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )} =C_2 e^{\frac{1}{\lambda } \varphi ^*(Nm \lambda )} \end{aligned}$$

where $C_2=C_1|K|$ and $|K|=\int _{K} 1 dx$ denotes the Lebesgue measure of K, hence concluding that $\mathcal {E}^{\omega }(B(x_0,{\delta });P).$ $\square$

Next, as an immediate consequence of the previous theorem we prove that an iterate of a constant coefficient hypoelliptic operator is a ultradifferentiable function.

Corollary 6.3

Let $\omega$ be a weight function, $\Omega \subset \mathbb {R}^n$ be an open set and P(D) be a constant coefficient hypoelliptic linear partial differential operator of order m. Then, denoting $\sigma (t)=\omega (t^{\rho })$, it follows that $\mathcal {E}^{\omega }(\Omega ;P)\subset {\mathcal {E}}^{\sigma }(\Omega )$.

Proof

If $u \in \mathcal {E}^{\omega }(\Omega ;P)$ then, using Theorem 6.2 it follows that u satisfies (6.1). Using Theorem 5.6, for $N=0$, it follows that $u \in \mathcal {E}^{\sigma }.$ $\square$

The next result is the so called Denjoy-Carleman Kotake-Narasimhan theorem for constant coefficients operator and the proof given here is different from the one in [7].

Corollary 6.4

Let $\omega$ be a weight function and $\Omega \subset \mathbb {R}^n$ be an open set. If P(D) is a constant coefficient elliptic linear partial differential operator of order m, then $\mathcal {E}^{\omega }(\Omega ;P)\subset {\mathcal {E}}^{\omega }(\Omega )$.

Proof

Since P(D) is a constant coefficient elliptic linear partial differential operator it follows that $\rho =1$ in (1.8). Thus, the result now follows from Corollary 6.3. $\square$

Theorem 6.5

Let $\omega$ be a weight function and $\Omega \subset \mathbb {R}^n$ be an open set. If P(D) is a constant coefficient linear partial differential operator of order m (non necessarily hypoelliptic), then $\mathcal E^{\omega }(\Omega )\subset \mathcal {E}^{\omega }(\Omega ;P)$.

Proof

Considering $u \in \mathcal {E}^{\omega }(\Omega )$, for each $K \subset \Omega$ compact there exist $C,\lambda >0$ such that

$$\begin{aligned} |\partial ^{\alpha } u(x)| \le Ce^{\frac{1}{\lambda } \varphi ^*(\lambda |\alpha |)}, \end{aligned}$$

for each $x \in K$ and $\alpha \in \mathbb {N}_0^n.$ Moreover, denoting $P(D)= \sum _{|\alpha | \le m} a_{\alpha } D^{\alpha } u$ it follows that

$$\begin{aligned} \big \Vert [P(D)]^ju\big \Vert _{L^2(K)}&\le \sum _{|\alpha _1|\le m} |a_{\alpha _1}| \cdots \sum _{|\alpha _j|\le m} |a_{\alpha _j}|\Vert \partial ^{\alpha _1 + \cdots +\alpha _j} u\Vert _{L^2(K)} \\&\le \sum _{|\alpha _1|\le m} |a_{\alpha _1}| \dots \sum _{|\alpha _j|\le m} |a_{\alpha _j}| \,C \,{|K|}\, e^{\frac{1}{\lambda } \varphi ^*(\lambda |\alpha _1 + \cdots +\alpha _j|)}. \end{aligned}$$

where $|K|=\int _{K} 1 dx$ denotes the Lebesgue measure of K. Since $t \mapsto \frac{\varphi ^*(t)}{t}$ is increasing, by denoting $\displaystyle h=\sum _{|\alpha |\le m} |a_{\alpha }|$ it follows that

$$\begin{aligned} \Vert [P(D)]^ju\Vert _{L^2(K)}&\le h^{j} C \Vert K\Vert e^{\frac{1}{\lambda } \varphi ^*(\lambda jm)}. \end{aligned}$$

Therefore, using (2.5) there exist $C_1>0$ and $\lambda _*>0$ such that

$$\begin{aligned} \big \Vert [P(D)]^ju\big \Vert _{L^2(K)} \le C |K| \left( h^{1/m}\right) ^{mj} e^{\frac{1}{\lambda } \varphi ^*(\lambda jm)} \le C |K| e^{\frac{1}{\lambda _*} \varphi ^*(\lambda _* jm)}. \end{aligned}$$

Thus we conclude that $u \in \mathcal {E}^{\omega }(\Omega ;P).$ $\square$

Remark 6.6

It is important to observe that using Corollary 6.4 and Theorem 6.5 we can prove that if $f \in \mathcal {E}^{\omega }(\Omega )$, P(D) is a constant coefficient elliptic linear partial differential operator of order m and $u \in L^2_{loc}(\Omega )$ is a solution of the equation

$$\begin{aligned} P(D)u=f, \quad \text{ in } \quad \Omega \end{aligned}$$

(6.12)

then, $u \in \mathcal {E}^{\omega }(\Omega ).$

Analogously, we can prove that if P(D) is a non-elliptic operator but it is a constant coefficient hypoelliptic linear partial differential operator of order m satisfying (1.8) and $u \in L^{2}_{loc}(\Omega )$ satisfies (6.12) then $u \in {\mathcal {E}}^{\sigma }(\Omega )$ (where $\sigma (t)=\omega (t^{\rho })$).

Appendix: Some Properties for a Constant Coefficient Hypoelliptic Linear Operator

Throughout this section we will consider $x_0 \in \mathbb {R}^n$, $0<R<1$, $m>0$ and a constant coefficient hypoelliptic linear operator P(D) of order m, defined in $B(x_0,R).$ Also, we will denote $\partial ^{\alpha }P=P^{(\alpha )}$ and $P^{(\alpha )}(D)f=P^{(\alpha )}f$, for each $\alpha \in \mathbb {N}_0^n$ and $f \in C^{\infty }(B(x_0,R)).$ Moreover, in order to simplify the notation we will denote $\Vert f\Vert _{r}=\Vert f\Vert _{L^2(B(x_0,r))}< \infty ,$ for each $0<r<R$. In the proof of the next theorem we shall use [20, inequality (4.5)’]. To be more precise, there exist $C,\gamma >0$ such that,

$$\begin{aligned} \epsilon ^{-|\alpha |}\Vert P^{(\alpha ) } \phi \Vert _{R}\le C\left( \Vert P\phi \Vert _R+(1+\epsilon ^{-\gamma }) \Vert \phi \Vert _R\right) , \end{aligned}$$

(A.1)

for each $\epsilon >0$, $\phi \in C_c^{m}(B(x_0,R))$ and $\alpha \in \mathbb {N}_0^n$ (such that $0<|\alpha |\le m$). Furthermore, following [20, Lemma 4.1] for each $\epsilon ,\epsilon _1>0$ such that $R > \epsilon _1+\epsilon$ there exists $\phi _{(\epsilon _1,\epsilon )} \in C_c^{\infty } (B(x_0,R-\epsilon ))$ such that $\phi _{\epsilon _1,\epsilon } \equiv 1$ in $B(x_0,R-\epsilon _1-\epsilon )$ and there exists $C>0$ dependent of $R>0$ but not on ${(\epsilon _1,\epsilon )}$ satisfying,

$$\begin{aligned} \Vert \partial ^{\alpha } \phi _{\epsilon _1,\epsilon }\Vert _ {L^\infty } \le C \epsilon _1^{-|\alpha |}, \end{aligned}$$

(A.2)

for $|\alpha | \le m.$ We will use the following notation,

$$\begin{aligned} N_{k,r}(f)=\sup _{{r>}\delta >0} \delta ^{k} \Vert f\Vert _{r-\delta }, \end{aligned}$$

for $r\le R,$ $f \in L^2(B(x_0,r))$ and $k \in \mathbb {R}.$

Lemma A.1

Let $u \in C^{m}(B(x_0,R))$ and $\gamma >0.$ Then there exists $D>0$ (dependent of $\gamma ,P,R$) such that,

$$\begin{aligned} N_{\gamma -|\alpha |,R-\delta _1} (P^{(\alpha )}u) \le D [N_{\gamma ,R-\delta _1}(Pu)+\Vert u\Vert _{R-\delta _1}], \end{aligned}$$

for each ${R>} \delta _1>0$ and $\alpha \in \mathbb {N}_0^n,$ where $|\alpha |\le m.$

Proof

Consider arbitrary $0<\!\delta$ and $\delta _1$ such that $\delta _1+\delta <R.$ By (A.2) there exists $\phi _{\delta ,{\delta _1}}\!\in C_c^{\infty } \left( B(x_0,R-\delta _1-\tfrac{\delta }{2})\right) ,$ $\phi _{\delta ,\delta _1} \equiv 1$ in $B(x_0,R-\delta _1-\delta )$ and

$$\begin{aligned} \Vert \partial ^{\alpha } \phi _{\delta ,\delta _1}\Vert _{L^\infty } \le C\left( \frac{\delta }{2} \right) ^{-|\alpha |}, \end{aligned}$$

for each $\alpha \in \mathbb {N}_0^n$ such that $|\alpha |\le m$. Moreover, for each $\epsilon >0$ and $\beta \in \mathbb {N}_0^n$ such that $|\beta |\le m$ it follows that,

$$\begin{aligned} \epsilon ^{-|\beta |} \Vert P^{(\beta )} u\Vert _{R-\delta _1-\delta }= \epsilon ^{-|\beta |} \Vert P^{(\beta )} (\phi _{\delta ,\delta _1}u)\Vert _{R-\delta _1-\delta } \le \epsilon ^{-|\beta |} \Vert P^{(\beta )} (\phi _{\delta ,\delta _1}u)\Vert _{R}. \end{aligned}$$

Using (A.1) it follows that,

$$\begin{aligned} \epsilon ^{-|\beta |} \Vert P^{(\beta )} u\Vert _{R-\delta _1-\delta }\le C(\Vert P(\phi _{\delta ,\delta _1} u)\Vert _R+(1+\epsilon ^{-\gamma }) \Vert \phi _{\delta ,\delta _1} u\Vert _R). \end{aligned}$$

Next we recall that, $P(D)(fg)=\sum _{|\alpha |\le m} \frac{1}{\alpha !} P^{(\alpha )}g\,\times \, \partial ^{\alpha } f$ for $f,g\in C^{\infty }$. Hence,

$$\begin{aligned} \epsilon ^{-|\beta |} \Vert P^{(\beta )} u\Vert _{R-\delta _1-\delta }&\le C\left( \left\| \sum _{|\alpha |\le m} \frac{1}{\alpha !} P^{(\alpha )}u\,\, \partial ^{\alpha } \phi _{\delta ,\delta _1} \right\| _R+(1+\epsilon ^{-\gamma }) \Vert \phi _{\delta ,\delta _1} \Vert _{\infty } \Vert u\Vert _{R-\delta _1-\frac{\delta }{2}}\right) \\&\le C^2\left( \sum _{|\alpha |\le m} \frac{\delta ^{-|\alpha | }2^{|\alpha |}}{\alpha !} \left\| P^{(\alpha )}u\right\| _{R-\delta _1-\frac{\delta }{2}} +(1+\epsilon ^{-\gamma }) \Vert u\Vert _{R-\delta _1-\frac{\delta }{2}}\right) \\&\le C^2\left( \sum _{|\alpha |\le m}\left( \frac{\delta }{2}\right) ^{-\gamma } \left( \frac{\delta }{2}\right) ^{\gamma -|\alpha |} \left\| P^{(\alpha )}u\right\| _{R-\delta _1-\frac{\delta }{2}} +(1+\epsilon ^{-\gamma }) \Vert u\Vert _{R-\delta _1}\right) \\&\le C^2\left( \sum _{|\alpha |\le m}\left( \frac{\delta }{2}\right) ^{-\gamma } N_{\gamma -|\alpha |,R-\delta _1}(P^{(\alpha )}u) +(1+\epsilon ^{-\gamma }) \Vert u\Vert _{R-\delta _1}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} \delta ^{\gamma }\epsilon ^{-|\beta |} \Vert P^{(\beta )} u\Vert _{R-\delta _1-\delta }&\le C^2\left( \sum _{|\alpha |\le m}2^{\gamma } N_{\gamma -|\alpha |,R-\delta _1}(P^{(\alpha )}u) +\left( \delta ^{\gamma }+\left( \frac{\delta }{\epsilon }\right) ^{\gamma }\right) \Vert u\Vert _{R-\delta _1}\right) {.} \end{aligned}$$

Next, considering an arbitrary $\chi >0$ and defining $\epsilon =\frac{\delta }{\chi }$ it follows that,

$$\begin{aligned} \chi ^{|\beta |} \delta ^{\gamma -|\beta |} \Vert P^{(\beta )} u\Vert _{R-\delta _1-\delta }&\le C^2\left( \sum _{|\alpha |\le m}2^{\gamma } N_{\gamma -|\alpha |,R-\delta _1}(P^{(\alpha )}u) +\left( \delta ^{\gamma }+ \chi ^{\gamma }\right) \Vert u\Vert _{R-\delta _1}\right) . \end{aligned}$$

Since $0<\delta<R<1$ and $\gamma >0$ we have $\delta ^{\gamma }<1$. Considering the supremum in $\delta >0$ in the above inequality, we have

$$\begin{aligned} \chi ^{|\beta |} N_{\gamma -|\beta |;R-\delta _1} \left( P^{(\beta )} u\right)&\le C^2\left( \sum _{|\alpha |\le m}2^{\gamma } N_{\gamma -|\alpha |,R-\delta _1}(P^{(\alpha )}u) +\left( 1+\chi ^{\gamma }\right) \Vert u\Vert _{R-\delta _1}\right) , \end{aligned}$$

for each $\beta \in \mathbb {N}_0^n$ such that $0<|\beta |\le m$. Summing in $\beta$ the last inequality we see that, there exists $C_1>0$ such that

$$\begin{aligned} \sum _{0<|\alpha |\le m}\chi ^{|\alpha |} N_{\gamma -|\alpha |;R-\delta _1} \left( P^{(\alpha )} u\right)&\le C_1\left( \sum _{|\alpha |\le m}2^{\gamma } N_{\gamma -|\alpha |,R-\delta _1}(P^{(\alpha )}u) +\left( 1+\chi ^{\gamma }\right) \Vert u\Vert _{R-\delta _1}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} \sum _{0<|\alpha |\le m} \left( \chi ^{|\alpha |} - C_1 2^{\gamma }\right) N_{\gamma -|\alpha |;R-\delta _1} \left( P^{(\alpha )} u\right)&\le C_1\left( 2^{\gamma } N_{\gamma ,R-\delta _1}(u) +\left( 1+\chi ^{\gamma }\right) \Vert u\Vert _{R-\delta _1}\right) . \end{aligned}$$

Choosing $\chi >1$, such that $\chi -C_1 2^{\gamma }>1,$ it follows that,

$$\begin{aligned} N_{\gamma -|\alpha |;R-\delta _1} \left( P^{(\alpha )} u\right)&\le C_1\left( 2^{\gamma } N_{\gamma ,R-\delta _1}(u) +\left( 1+\chi ^{\gamma }\right) \Vert u\Vert _{R-\delta _1}\right) , \end{aligned}$$

for each $\alpha \in \mathbb {N}_0^n$ such that $0<|\alpha |\le m.$ Therefore, denoting $C_2=\max \{2^{\gamma };(1+\chi ^{\gamma })\}$ and $D=C_1 C_2,$ it follows that,

$$\begin{aligned} N_{\gamma -|\alpha |;R-\delta _1} \left( P^{(\alpha )} u\right)&\le D\left( N_{\gamma ,R-\delta _1}(u) + \Vert u\Vert _{R-\delta _1}\right) , \end{aligned}$$

for each $\alpha \in \mathbb {N}_0^n$ such that $0<|\alpha |\le m.$ $\square$

Corollary 6.7

Let $u \in C^{m}(B(x_0,R)).$ There exists $C_P>0$ such that,

$$\begin{aligned} \Vert P^{(\alpha )}u\Vert _r \le C_P \epsilon ^{|\alpha |} \left( \Vert Pu\Vert _{r+\epsilon } + \Vert u\Vert _{r+\epsilon } \right) , \end{aligned}$$

for each $\epsilon ,r>0$ (such that $r+\epsilon \le R<1$) and $\alpha \in \mathbb {N}_0^n$ where $|\alpha |<m.$ Let us recall the notation $\Vert u\Vert _{r+\epsilon }=\Vert u\Vert _{L^2(B(x_0,r+\epsilon ))}.$

Proof

Using Lemma A.1 it follows that,

$$\begin{aligned} \delta ^{\gamma -|\alpha |} \Vert P^{(\alpha )}u \Vert _{R-\delta _1-\delta }&\le D\left[ \sup _{0<\eta <R-\delta _1} \eta ^{\gamma } \Vert Pu\Vert _{R-\delta _1-\eta } + \Vert u\Vert _{R-\delta _1} \right] \le D\left( R^{\gamma } \Vert Pu\Vert _{R-\delta _1} + \Vert u\Vert _{R-\delta _1} \right) \end{aligned}$$

for each $\delta ,\delta _1>0$ such that $\delta +\delta _1<R$ and $\alpha \in \mathbb {N}_0^n$, such that $0<|\alpha |\le m.$ Therefore, for each $0<r<R$ and $0<\epsilon \le R- r$ it follows that

$$\begin{aligned} \epsilon ^{\gamma -|\alpha |} \Vert P^{(\alpha )}u\Vert _r= \epsilon ^{\gamma -|\alpha |} \Vert P^{(\alpha )}u\Vert _{R-(R-r-\epsilon )-\epsilon } \le D \left( R^{\gamma } \Vert Pu\Vert _{r+\epsilon } + \Vert u\Vert _{r+\epsilon } \right) . \end{aligned}$$

Hence, using that $0<R<1$

$$\begin{aligned} \Vert P^{(\alpha )}u\Vert _r\le \epsilon ^{-\gamma } D \epsilon ^{|\alpha |} \left( \Vert Pu\Vert _{r+\epsilon } + \Vert u\Vert _{r+\epsilon } \right) \le C_P\epsilon ^{|\alpha |} \left( \Vert Pu\Vert _{r+\epsilon } + \Vert u\Vert _{r+\epsilon } \right) , \end{aligned}$$

where $C_P=\epsilon ^{\gamma } D$. $\square$